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Characterization of QUBO reformulations for the maximum
k-colorable subgraph problem
Rodolfo Quintero∗David Bernal†Tam´as Terlaky‡Luis F. Zuluaga§
January 26, 2021
Abstract
Quantum devices can be used to solve constrained combinatorial optimization (COPT) prob-
lems thanks to the use of penalization methods to embed the COPT problem’s constraints in
its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation
of the COPT. However, the particular way in which this penalization is carried out, affects the
value of the penalty parameters, as well as the number of additional binary variables that are
needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect
the ability of quantum computers to efficiently solve these constrained COPT problems. This
efficiency is key towards the goal of using quantum computers to solve constrained COPT prob-
lems more efficiently than with classical computers. Along these lines, we consider an important
constrained COPT problem; namely, the maximum k-colorable subgraph (MkCS) problem, in
which the aim is to find an induced k-colorable subgraph with maximum cardinality in a given
graph. This problem arises in channel assignment in spectrum sharing networks, VLSI design,
human genetic research, and cybersecurity. We derive two QUBO reformulations for the MkCS
problem, and fully characterize the range of the penalty parameters that can be used in the
QUBO reformulations. Further, one of the QUBO reformulations of the MkCS problem is ob-
tained without the need to introduce additional binary variables. To illustrate the benefits of
obtaining and characterizing these QUBO reformulations, we benchmark different QUBO refor-
mulations of the MkCS problem by performing numerical tests on D-Wave’s quantum annealing
devices. These tests also illustrate the numerical power gained by using the latest D-Wave’s
quantum annealing device.
1 Introduction
Quantum computing (QC) harnesses the properties physical systems described by quantum me-
chanics (e.g., subatomic particles) to perform computations in a fundamentally different way
than classical computing [55]. It is widely established that QC can, in the future, revolution-
ize the way we perform and think about computation, and be the backbone of thrilling new
technologies and products [15,44,55].
In particular, QC has the potential to radically transform our capability to solve difficult
optimization problems for which no traditional numerical or theoretical efficient solution al-
gorithms are known to exist [45]. This is particularly the case for combinatorial optimization
(COPT) problems; that is, optimization problems that are formulated with the use of discrete
∗Industrial and Systems Engineering, Lehigh University, USA roq219@lehigh.edu
†Department of Chemical Engineering, Carnegie Mellon University, USA, bernalde@cmu.edu
‡Department of Industrial and Systems Engineering, Lehigh University, USA, terlaky@lehigh.edu
§Department of Industrial and Systems Engineering, Lehigh University, USA, luis.zuluaga@lehigh.edu
1
arXiv:2101.09462v1 [quant-ph] 23 Jan 2021
(e.g., binary) decision variables [16]. A large number of COPT problems are known to be NP-
Hard [see, e.g., 34]; that is, there is no known polynomial-time algorithm that can be used to
solve them. A very representative problem in this class of COPT NP-Hard problems is the
Ising model [see, e.g., 9,14,56]. Since its inception, the Ising model has been used to address
problems arising in different physical systems (e.g., magnetism, lattice gas, spin glasses), as well
as in neuroscience and socio-economics.
The Ising model belongs to the class of quadratically unconstrained binary optimization
(QUBO) problems [see,e.g., 48]. Moreover, both quantum annealing devices [see, e.g., 13,33,42],
and algorithms (such as the quantum approximate optimization algorithm (QAOA)) for gate-
based quantum computers [see, e.g., 23,64] are able to address the solution of QUBO problems.
This allows the use of quantum technology to solve problems such as the Ising model and
the max-cut problem, which has a natural QUBO reformulation [see, e.g., 20,36]. Moreover,
quantum technology can be used to solve a broader class of constrained COPT problems that
do not have a natural QUBO reformulation. This is due to the fact that penalization methods
can be used to embed the COPT problem’s constraints in its objective to obtain a QUBO
reformulation of the problem.
For some COPT feasibility problems (i.e., without an objective) that can be formulated
using linear equality constraints, the desired QUBO reformulation can be obtained using any
positive penalty parameter (to penalize the constraints’ violations). For example, consider the
QUBO reformulations of the number partitioning problem [42,46], the graph isomorphism
problem [10], the exact cover problem [42], and some planning problems [51], to name a few.
However, when the COPT problem formulation requires (or uses) nonlinear constraints and/or
an objective function, the desired QUBO reformulation is only guaranteed to be obtained for
values of the penalty parameter(s) that are larger than a known, and potentially large, lower
bound. For example, consider the QUBO reformulations for the maximum clique problem [42],
the traveling salesman problem [42,46], and the minimax matching problem [42]. Worst, in some
cases, the desired QUBO reformulation is only guaranteed to be obtained for an unknown large
enough value of the penalty parameter(s). For example, consider the QUBO reformulations
of the job shop scheduling problem [60], the de-conflicting optimal trajectories problem [57],
the traveling salesman problem with time windows [49], and some of the problems discussed
in [26]. Additionally, when the COPT problem formulation requires (or uses) linear inequality
constraints, a potentially large number of auxiliary (i.e., slack) binary variables need to be
introduced to obtain the desired QUBO reformulation. For example, consider the maximum
clique QUBO reformulation provided in [42], and the COPT problems considered in [63].
The fact that large (or unknowingly large) penalty parameters, and additional binary vari-
ables might be needed to obtain the desired QUBO reformulation can hinder the ability of
quantum computers to more efficiently solve COPT problems [see, e.g., 25,54,63]. As the
results in [28] highlight, this efficiency is key towards the goal of using noisy intermediate scale
quantum (NISQ) devices to solve COPT problems more efficiently than with classical comput-
ers. Not surprisingly, recent articles look beyond obtaining QUBO reformulations of COPT
problems such as the graph isomorphism problem as well as tree and cycle elimination prob-
lems, to look for improved QUBO reformulations of these problems for NISQ devices [see, e.g.,
10,25,31,61,62]. That is, QUBO reformulations that are tailored to be more efficiently used
in NISQ devices.
Along these lines, we consider an important COPT problem; namely, the maximum k-
colorable subgraph (MkCS) problem [see, e.g., 38], in which the aim is to find an induced k-
colorable subgraph with maximum cardinality in a given graph. This problem arises in channel
assignment in spectrum sharing networks (e.g., Wi-Fi or cellular) [29,58], VLSI design [24],
human genetic research [24,40], telecommunications [41], and cybersecurity [4].
We derive two QUBO reformulations of the MkCS problem. The first one is obtained from
the standard formulation of the MkCS problem in which all the constraints are linear, except for
the binary variable constraints. This QUBO reformulation is an improved version of the QUBO
reformulation that would be obtained by using the QUBO reformulation approach of Lasserre
[39] for this “linear” formulation of the MkCS. The reason for this is that we characterize the
2
minimum penalization coefficients that can be used to guarantee that the desired QUBO prob-
lem, obtained by penalizing the problem’s linear constraints violations, is indeed equivalent to
the original problem. Furthermore, we characterize the equivalence of the QUBO reformulation
not only in terms of the objective value, but also in terms of the optimal solution obtained
from this QUBO reformulation. In particular, we find that when the minimal values of the
penalization coefficients are used, the QUBO reformulation is equivalent to the MkCS in terms
of the problems’ objectives, but not in terms of the problems’ optimal solutions. However, we
show that in this case, the QUBO reformulation’s optimal solution can be used, in a simple way,
to obtain the MkCS problem’s optimal solution. In what follows, we will refer to this QUBO
reformulation of the MkCS problem as the linear-based QUBO reformulation.
The second QUBO reformulation of the MkCS problem is obtained from a formulation of
the MkCS problem in which all the linear constraints are first formulated as nonlinear equality
constraints. Analogous to the results obtained for the linear-based QUBO reformulation of
the MkCS problem, we derive a nonlinear-based QUBO reformulation of the MkCS problem.
Then, we characterize the minimum penalizations coefficients that can be used to guarantee
that the desired nonlinear-based QUBO problem, obtained by penalizing the problem’s linear
constraints violations, is indeed equivalent to the original problem. Furthermore, we characterize
the equivalence of the nonlinear-based QUBO reformulation not only in terms of the objective
value, but also in terms of the optimal solution obtained from this nonlinear-based QUBO
reformulation. In particular, we find that when the minimal values of the penalization coefficients
are used, the nonlinear-based QUBO reformulation is equivalent to the MkCS in terms of the
problems’ objectives, but not in terms of the problems’ optimal solutions. However, we show
that in this case, the nonlinear-based QUBO reformulation’s optimal solution can be used, in a
simple way, to obtain the MkCS problem’s optimal solution. This latter result extends the work
done in characterizations of QUBO reformulations of the stable set problem [1,8,30], which is
equivalent to the MkCS problem when k= 1. The nonlinear-based QUBO reformulation of the
MkCS problem is a substantial improvement over the linear-based QUBO reformulation of the
MkCS problem, in significant part, because the former QUBO does not need the addition of
any auxiliary (i.e., slack) binary variables beyond the ones that define the original problem’s
formulation.
To illustrate the benefits of obtaining and characterizing these QUBO reformulations, we
benchmark different QUBO reformulations of the MkCS problem using a quantum annealing
device, and in particular, we look at how embedding requirements and theoretical and numerical
convergence rates change depending on the QUBO reformulation being used, as well as the
parameters with which is used.
The rest of the article is organized as follows. In Section 2, we present some relevant discus-
sion to motivate our work, as well as results about QUBO reformulations for COPT problems.
In Section 3, we formally present the MkCS problem and two associated QUBO reformulations.
The first one, in Section 3.1, is based on a “linear” (modulo the binary variable constraints)
formulation of the MkCS problem. The second one, in Section 3.2, is based on a “nonlinear”
(beyond the binary variable constraints) formulation of the MkCS problem. In Section 4, we
benchmark these two QUBO reformulation by performing numerical tests on D-Wave’s quantum
annealing devices. We also illustrate the numerical power gained by using the latest D-Wave’s
quantum annealing devices. In Section 5, we finish with some concluding remarks.
2 Preliminaries
Formally, given a set of nbinary decision variables x∈ {0,1}n(or x∈ {−1,1}n) when ap-
propriate), a vector f∈Rn, and a matrix Q∈ Sn, where Snis the set of symmetric matrices
in Rn×n, a quadratically unconstrained binary optimization (QUBO) problem is the problem of
finding [see, e.g., 8,48]:
z∗= min x|Qx +f|x
s.t. x ∈ {0,1}n.(QUBO)
3
It is well-known that the Ising model belongs to the class of QUBO problems (using {−1,1}
binary variables) [see, e.g., 25,42]. Moreover, other distinguished NP-Hard COPT problems can
be naturally formulated, or easily reformulated as a QUBO problem. Foremost among this type
of problems is the max-cut problem [see, e.g., 27], which arises in multiple important applications
in science and engineering [see, e.g., 50, Sec. 6]. Given an undirected graph G(V , E), the aim
in the max-cut problem is to find a subset of nodes (or cut) S⊆V, such that the cardinality
of the set of edges in Ebetween the nodes in Sand Sc:= V\Sis maximized. The max-cut
problem can be naturally formulated (disregarding objective constants) as a QUBO problem
(using {−1,1}binary variables) by letting Q=A,f= 0, where A∈RV×Vis the node-to-node
adjacency matrix of G(V, E ), or by setting Q=−diag(Ae)+2Aand f= 0 (using {0,1}binary
variables).
Thanks to the QUBO reformulation of the max-cut problem, the ability of quantum com-
puters to solve the max-cut problem has been widely studied in the literature. For example,
consider the use QAOA algorithms in [17,23,64], and of quantum annealing devices in [36,37]
to solve instances of the max-cut problem. Furthermore, QUBO reformulations can be obtained
for a broader class of COPT problems that do not have a natural QUBO reformulation. This
is done by using penalization methods to embed the COPT problem’s constraints in its objec-
tive [see, e.g., 10,25,26,42,46,51,57,60,63, to name just a few]. This approach clearly
broadens the class of COPT problems that can be addressed with NISQ devices. However, the
efficacy of NISQ devices to solve this broader class of COPT problems can be highly affected
by the way in which the corresponding QUBO reformulation is obtained. This is because the
performance of NISQ devices is highly affected by the number of qubits and the coefficients that
are required to encode a QUBO [see, e.g., 10,25,31].
To illustrate this fact, consider the problem of obtaining a QUBO reformulation for the
maximum clique problem. Given an undirected graph G(V , E), the aim in the maximum clique
problem is to find the set of nodes S⊆Vwith the highest cardinality such that the graph induced
by Sis a clique; that is, a complete subgraph [see, e.g., 6]. The cardinality of the largest induced
clique of Gis referred to as the clique number χ(G). Lucas [42, Sec. 2.3] obtains a QUBO
reformulation for the maximum clique problem by first noticing that G(V, E ) contains a clique
of size K∈ {2,...,|V|} (i.e., w.l.o.g. assume |E| ≥ 1) if and only if there is x∈ {0,1}|V|such
that P|V|
i=1 xi=K, and P(i,j)∈Exixj=1
2K(K−1). Thus, the maximum clique problem can
be formulated as χ(G) = max{K∈ {2,...,|V|} :P|V|
i=1 xi=K, P(i,j)∈Exixj=1
2K(K−1), x ∈
{0,1}|V|}. Furthermore, Lucas [42, Sec. 2.3] shows that this latter problem can be reformulated
as the following QUBO.
χ(G) = min −
|V|
X
i=1
xi+ (∆ + 2) 1−
∆
X
k=2
yk!2
+ (∆ + 2)
∆
X
k=2
kyk−
|V|
X
i=1
xi
2
+
1
2 ∆
X
k=2
kyk! −1 +
∆
X
k=2
kyk!−X
(i,j)∈E
xixj
s.t. x ∈ {0,1}|V|, yk∈ {0,1}, k = 2,...,∆,
(1)
where ∆ is the degree of G(V, E ), and the auxiliary variable yk= 1 if χ(G) = kand yk= 0
otherwise for k= 2,...,∆. Note that the QUBO problem (1) uses |V|+ ∆ logical qubits and
coefficients [63, Section 1.2] that belong to the range [−2∆(∆ + 2),2∆3+ 3∆(∆ −1) + 4] (after
disregarding constant terms and appropriately replacing xi→x2
i,i= 1,...,|V|,yk→y2
k,
k= 2,...,∆ in the objective of (1) to make it a homogenous quadratic). The performance of
NISQ devices on solving QUBO problems is negatively affected by the use of a larger number
of logical qubits and larger coefficients [see, e.g., 10,25,26,31,63]. In this context, it is natural
to ask if there are improved [see, e.g., 10,25,31,61,62] QUBO reformulation for the maximum
clique problem. For example, notice that by slightly changing the definition and number of the
auxiliary variables in (1), the range of the coefficients used in (1) can be substantially reduced.
Namely, let y∈ {0,1}∆be defined by P∆
k=1 yk=Kif χ(G) = Kfor K∈ {1,...,∆}. Then the
4
maximum clique problem is equivalent to:
χ(G) = min −
|V|
X
i=1
xi+ (∆ + 2)
∆
X
k=1
yk−
|V|
X
i=1
xi
2
+
1
2 ∆
X
k=1
yk! −1 +
∆
X
k=1
yk!−X
(i,j)∈E
xixj
s.t. x ∈ {0,1}|V|, y ∈ {0,1}∆.
(2)
Note that the QUBO problem (2) uses coefficients that belong to a much smaller range [−2(∆+
2),4(∆+ 2) +1] than the range of coefficients used in the QUBO problem (1) (after disregarding
constant terms and appropriately replacing xi→x2
i,i= 1,...,|V|,yk→y2
k,k= 1,...,∆
in the objective of (1) to make it an homogenous quadratic). However, a much better QUBO
formulation for the maximum clique problem can be obtained by using the fact that χ(G) =
α(Gc) [see, e.g., 6], where for a graph G(V , E), Gc=G(V, E c) is the complement of G, and
α(G) stands for the stable set number of the graph G[see, e.g., 30]; that is, the size of the largest
cardinality set S⊆V, such that there are no edges between the nodes in S. This fact can be
used to show that (see, e.g., [10, Thm. 6] or [6, Thm. 2.3], among others)
χ(G) = α(Gc) = min
−
|V|
X
i=1
xi+ 2 X
(i,j)6∈E
xixj:x∈ {0,1}|V|
(3)
Note that the QUBO problem (3) uses |V|logical qubits and coefficients that belong to the range
{−1,2}. Thus, in terms of number of logical qubits and range of the coefficients used in the
QUBO reformulation, (3) improves both (2) and (1). It is worth pointing out that the QUBO
reformulation (3) has been stated in numerous articles [see, e.g., 1,6,12,67, to name a few].
Moreover, it is well known that the range of the coefficients in (3) can be further reduced to
{−1,1}. Namely, it has been proved (or stated) in numerous articles [see, e.g., 1,8,30,46,48,65]
that
χ(G) = α(Gc) = min
−
|V|
X
i=1
xi+X
(i,j)6∈E
xixj:x∈ {0,1}|V|
(4)
There is, however, a caveat in the QUBO reformulation (4). For any x∈Rn, let supp(x) =
{i∈ {1, . . . , n}:xi6= 0}. Unlike for (1)–(3), given x∗∈arg min{(4)}, supp(x∗) might not
be a clique on G(nor an independent set in Gc). That is, while the QUBO problems (1)–(3)
are equivalent to the maximum clique problem in terms of both objective value and (loosely
speaking) optimal solution, in general, the QUBO problem (4) is equivalent to the maximum
clique problem only in terms of objective value. This important topic will be revisited and
discussed in detail in Section 3.2.
Along these lines, in what follows, we consider the problem of obtaining not only a QUBO
reformulation, but improved QUBO reformulation of a keystone COPT problem; namely, the
maximum k-colorable subgraph (MkCS) problem [see, e.g., 38].
3 The k-subgraph coloring problem
Let k≥1 colors and a graph G= (V, E ) on nvertices be given. A subgraph Hof Gis k-
colorable if we can assign to each vertex of Ha color such that no two adjacent vertices in H
have the same color. The maximum k-colorable subgraph problem (MkCS) aims at finding a
k-colorable subgraph Hof Gwith maximum cardinality. To model this problem, notice that
any k-coloring of a subgraph of Gcan be encoded in the following way. For any i∈[n] (where
5
for any t∈N, [t] := {1, . . . , t}) and r∈[k], let
xir =(1,if vertex i∈[n] is colored with color r∈[k],
0,otherwise.(5)
Then, x∈ {0,1}n×kdefines a k-coloring of a subgraph of Gif and only
xir +xjr ≤1,for all (i, j )∈E, r ∈[k],
X
r∈[k]
xir ≤1,for all i∈[n].(6)
Then, the MkCS can be formulated as [see, e.g., 38]:
αk(G) := max
x∈{0,1}n×kX
i∈[n],r∈[k]
xir
s.t. xir +xjr ≤1,for all (i, j )∈E, r ∈[k],
X
r∈[k]
xir ≤1,for all i∈[n].
(7)
The MkCS problem falls into the class of NP-complete problems [66]. Moreover, even approx-
imating this problem is known to be NP-hard [43]. For k= 1, the MkCS is equivalent to the
maximum stable set problem (i.e., α1(G) = α(G)) that has been widely and thoroughly studied
in the literature; and in particular, in the quantum computing literature [see, e.g., 12,46,65].
The cases k= 2, which is also referred to as the maximum bipartite subgraph problem, and k > 2
are considered significantly less in the literature [see 38, for details]. However, as mentioned ear-
lier, the MkCS problem arises in channel assignment in spectrum sharing networks (e.g., Wi-Fi
or cellular) [29,58], VLSI design [24], human genetic research [24,40], telecommunications [41],
and cybersecurity [4]. Thus, a range of approaches have been studied in the literature to address
the solution of the MkCS problem, for example, using semidefintie optimization techniques [see,
e.g., 38,59] or integer programming techniques [see, e.g., 10,11,32].
Next, we obtain and characterize QUBO reformulations for the MkCS problem that allow to
address its solution using quantum technology. Before presenting these results, let us mention
some additional facts about the MkCS problem that will be relevant to the discussion in what
follows.
Notice that a MkCS Hof G(V, E ) can be recovered from any x∗∈arg max{αk(G)}; that
is, H:= G(VH, EH), where VH={i∈[n] : x∗
ir >0 for some r∈[k]},EH:= {(i, j)∈E:
i, j ∈VH}, and the coloring of the vertices is obtained by coloring vertex i∈VHwith color
r∈[k] if and only if x∗
ir = 1. Furthermore, given ˜x∈ {0,1}n×k, it is very simple to obtain a
feasible solution x0∈ {0,1}n×kfor the MkCS problem by sequentially dropping color r0∈[k]
from vertex i0∈[n]; that is, setting ˜xi0r0= 0, if ˜xi0r0= 1 and there exists (i0, j)∈Esuch that
˜xi0r0+ ˜xjr0>1 or Pr6=r0˜xi0r≥1. This simple fact is formally stated in Algorithm 1, in a
particular form that will be helpful in stating some of the QUBO characterization results that
follow.
3.1 Linear-based QUBO reformulation
Based on the formulation (7) of the MkCS problem in which all the constraints, except for
the binary variable constraints are linear, we can derive and characterize a linear-based QUBO
reformulation for the MkCS problem. For that purpose, let us first introduce some notation.
Given k≥1, a graph G= (V, E ) on nvertices, and x∈ {0,1}n×k,s∈ {0,1}|E|×k,t∈ {0,1}n,
let
H0(x) = X
i∈[n],r∈[k]
x2
ir,(8)
6
Algorithm 1 MkCS feasibility
1: Input k≥1, G(V, E ), |V|=n,x∈ {0,1}n×k
2: for i∈[n], (i, j)∈E,r∈[k]do
3: if xir +xjr >1then
4: xir →0
5: end if
6: end for
7: for i∈[n], r∈[k]do
8: if xir = 1 and Pp6=r∈[k]xip ≥1then
9: xir →0
10: end if
11: end for
12: Output x0:= xa feasible solution for the MkSC problem
and
Hl
1(x, s) = X
(i,j)∈E,r∈[k]
(xir +xjr +sij r −1)2,(9a)
Hl
2(x, t) = X
i∈[n]
X
r∈[k]
xir +ti−1
2
.(9b)
Furthermore, we define the following simple mappings. Given x∈ {0,1}n×kand i0∈[n], r0∈[k],
let the mapping Xi0r0(x) : {0,1}n×k→ {0,1}n×kbe defined by
xir →0 if i=i0, r =r0
xir otherwise , i ∈[n], r ∈[k].(10)
Note that Xi0r0(x) is a generalization of the mapping used on proofs regarding QUBO reformu-
lations of the stable set number problem (i.e., M1CS) [see, e.g., 1,8,30,48,65]. Here, however,
to deal with the general case k > 1, we need an additional mapping.
Given p={0,1},s∈ {0,1}|E|×k,t∈ {0,1}n, and (i0, j0)∈E,r0∈[k], let the mapping
Mp
i0j0,r0(s, t) : {0,1}|E|×k+n→ {0,1}|E|×k+nbe defined by
sijr →
1−si0jr0if i=i0, j 6=j0, r =r0
(1 −si0jr0)pif i=i0, j =j0, r =r0
sijr otherwise
,(i, j)∈E , r ∈[k],(11a)
ti→1−pif i=i0,
tiotherwise , i ∈[n].(11b)
With these definitions in hand, we can now obtain the desired linear-based QUBO reformu-
lation of the MkCS problem. For any c1, c2>0 define the QUBO problem:
Ql
c1,c2(k, G) := max Hl
c1,c2(x, s, t) := H0(x)−c1Hl
1(x, s)−c2Hl
2(x, t)
s.t. x ∈ {0,1}n×k, s ∈ {0,1}|E|×k, t ∈ {0,1}n.
(12)
Theorem 1 (linear-based QUBO reformulation of MkCS problem).Let k≥1and a graph
G= (V, E )on nvertices be given. Then, for any c1>1, c2>1,Ql
c1,c2(k, G) = αk(G), and if
˜x∈arg maxx{Ql
c1,c2(k, G)}then ˜x∈arg max{αk(G)}.
Proof. First, notice that ˜xis well defined and Ql
c1,c2(k, G) is attained as (12) is defined over a
compact feasible set. Also, notice that for any c1, c2>0 and any feasible solution x0∈ {0,1}n×k
7
for the MkCS problem (7) with objective value z(x0) := Pi∈[n],r∈[k]xir, one can construct a
feasible solution for (12); that is, x=x0,sij r = 1 −x0
ir −x0
jr , for all (i, j )∈E, r ∈[k], and
ti= 1 −Pr∈[k]x0
ir, with objective value Hl
c1,c2(x, s, t) = z(x0). Thus, if c1, c2>0, the QUBO
problem (12) is a relaxation of (7), and consequently Ql
c1,c2(k, G)≥αk(G). Thus, to prove the
result, it is enough to show that when c1, c2>1, one has that ˜xis a feasible solution for (7). By
contradiction, assume this is not the case and let c1, c2>1, (˜s, ˜
t) := arg max(s,t){Ql
c1,c2(k, G)}.
Then either: (1) there is at least an (i0, j0)∈Eand r0∈[k] such that ˜xi0r0+ ˜xj0r0>1; or (2)
there is at least an i0∈[n] and r0∈[k] such that ˜xi0r0= 1 and Pr6=r0∈[k]˜xi0r≥1.
For case (1), consider the feasible solution (x, s0, t0)∈ {0,1}n×k+|E|×k+nfor (12) obtained
from (˜x, ˜s, ˜
t) by letting (x, s, t) = (Xi0r0(˜x),M0
i0j0,r0(˜s, ˜
t)) (cf., (10), (11)). It then follows
from (8), (10), and the fact that ˜xi0r0= 1 that
H0(x) = H0(˜x)−1.(13)
Also, from (9a), (10), (11a), and the fact that ˜xi0r0= ˜xj0r0= 1, it follows that −Hl
1(x, s0) =
−Hl
1(˜x, ˜s) + P(i0,j6=j0)∈E4˜xj r0˜si0jr0+ (1 + ˜si0j0r0)2. Thus,
−Hl
1(x, s0)≥ −Hl
1(˜x, ˜s)+1.(14)
Further, from (9b), (10), (11b), and the fact that ˜xi0r0= 1, it follows that −Hl
2(x, t0) =
−Hl
2(˜x, ˜
t)+2˜
ti0Pr6=r0∈[k]˜xi0r+˜
t2
i0. Thus,
−Hl
2(x, t0)≥ −Hl
2(˜x, ˜
t).(15)
Using (13), (14), (15), it follows that Hl
c1,c2(x, s0, t0)≥Hl
c1,c2(˜x, ˜s, ˜
t)−1 + c1> Hl
c1,c2(˜x, ˜s, ˜
t) =
Ql
c1,c2(k, G), which contradicts the optimality of (˜x, ˜s, ˜
t) for (12).
We proceed analogously for case (2). Consider the feasible solution (x, s, t)∈ {0,1}n×k+|E|×k+n
for (12) obtained from (˜x, ˜s, ˜
t) by letting (x, s1, t1) = (Xi0r0(˜x),M1
i0,r0(˜s, ˜
t)) (cf., (10), (11)). It
then follows from (8), (10), and the fact that ˜xi0r0= 1 that (13) holds. Also, from (9a), (10), (11a),
and the fact that ˜xi0r0= 1, it follows that −Hl
1(x, s1) = −Hl
1(˜x, ˜s) + P(i0,j)∈E4˜xj r0˜si0jr0. Thus,
−Hl
1(x, s1)≥ −Hl
1(˜x, ˜s).(16)
Further, from (9b), (10), (11b), and the fact that ˜xi0r0= 1, Pr6=r0∈[k]˜xj0r≥1, it follows that
−Hl
2(x, t1) = −Hl
2(˜x, ˜
t) + 2(Pr6=r0∈[k]˜xi0r)(1 + ˜
ti0) + ˜
t2
i0−1. Thus,
−Hl
2(x, t1)≥ −Hl
2(˜x, ˜s)+1.(17)
Using (13), (16), (17), it follows that Hl
c1,c2(x, s1, t1)≥Hl
c1,c2(˜x, ˜s, ˜
t)−1 + c2> Hl
c1,c2(˜x, ˜s, ˜
t) =
Ql
c1,c2(k, G), which contradicts the optimality of (˜x, ˜s, ˜
t) for (12).
Therefore ˜xsatisfies that there is no (i0, j 0)∈Eand r0∈[k] such that ˜xi0r0+ ˜xj0r0>1, or
i0∈[n] and r0∈[k] such that Pr∈[k]˜xi0r>1. Therefore ˜xis a feasible solution of (7), which
finishes the proof.
It is worth to mention that, loosely speaking, the general form of the QUBO reformula-
tion (12) for the MkCS problem can be obtained by using the recent results of Lasserre [39,
Thm. 2.2]. Namely, one can use this result after reformulating the MkCS problem constraints
as equality constraints using the approach described in [39, Sec. 2.3]. Then, after reformulating
the problem using {1,−1}binary variables (instead of {0,1}binary variables), [39, Thm. 2.2]
can be used to obtain a QUBO reformulation of the MkCS problem. However, this reformula-
tion would require the use a penalty parameter with a value larger than nk (cf., with the values
of c1, c2in Theorem 1), and require the use of more auxiliary (i.e., slack) binary variables than
the ones used in Theorem 1. Thus, Theorem 1provides an improved QUBO reformulation of
the MkCS problem than the one that would be obtained using [39, Thm. 2.2].
Later, in Section 3.3, we will further characterize the QUBO reformulation (12) for the
MkCS. Next, however, we derive and characterize a QUBO reformulation for the MkCS in
which no auxiliary (i.e., slack) binary variables are needed.
8
3.2 Nonlinear QUBO reformulation
Next, we obtain an improved QUBO reformulation for the MkCS problem in terms of the
number of binary decision variables required in the QUBO reformulation, when compared with
the one provided and characterized in Section 3.1. For this purpose, first notice that for any
x∈ {0,1}n×k, the linear constraints in (6) are equivalent to the nonlinear constraints
xirxj r = 0,for all (i, j)∈E, r ∈[k],
xirxip = 0,for all i∈[n],(r, p 6=r)∈[k]×[k].(18)
Then, consistent with (18), given k≥1, a graph G= (V, E) on nvertices, and x∈ {0,1}n×k,
let
Hn
1(x) = X
(i,j)∈E,r∈[k]
xirxj r ,(19a)
Hn
2(x) = X
i∈[n]
X
r∈[k],p6=r∈[k]
xirxip
.(19b)
With these definitions in hand we can now obtain the desired nonlinear-based QUBO reformu-
lation of the MkCS problem. For any c1, c2>0 define the QUBO problem:
Qn
c1,c2(k, G) := max Hn
c1,c2(x) := H0(x)−c1Hn
1(x)−c2Hn
2(x)
s.t. x ∈ {0,1}n×k.
(20)
Theorem 2 (nonlinear-based QUBO reformulation of MkCS problem).Let k≥1and a graph
G= (V, E )on nvertices be given. Then, for any c1>1, c2>1,Qn
c1,c2(k, G) = αk(G), and if
˜x∈arg max{Qn
c1,c2(k, G)}then ˜x∈arg max{αk(G)}.
Proof. The proof is mostly analogous to the proof of Theorem 1. First, notice that ˜xis well
defined and Qn
c1,c2(k, G) is attained as (20) is defined over a compact feasible set. Also, notice
that for any c1, c2>0 and any feasible solution x0∈ {0,1}n×kfor the MkCS problem (7)
with objective value z(x0) := Pi∈[n],r∈[k]xir, one hast that x0is a feasible solution of (20)
with objective value Hn
c1,c2(x) = z(x0) (i.e., x0satisfies (18)). Thus, if c1, c2>0, the QUBO
problem (20) is a relaxation of (7), and consequently Qn
c1,c2(k, G)≥αk(G). Thus, to prove the
result, it is enough to show that when c1, c2>1, one has that ˜xis a feasible solution for (7). By
contradiction, assume this is not the case and let c1, c2>1. Then either: (1) there is at least an
(i0, j0)∈Eand r0∈[k] such that ˜xi0r0+ ˜xj0r0>1; or (2) there is at least an i0∈[n] and r0∈[k]
such that ˜xi0r0= 1 and Pr6=r0∈[k]˜xi0r≥1. Notice that in either case ˜xi0r0= 1. Now consider the
feasible solution x∈ {0,1}n×kfor (20) obtained from ˜xby letting x=Xi0r0(˜x) (cf., (10)). Notice
that from (8), (10), and the fact that ˜xi0r0= 1 one has that (13) holds. Also, from (19a), (10),
and the fact that ˜xi0r0= 1, it follows that −Hn
1(x) = −Hn
1(˜x) + P(i0,j6=j0)∈E˜xjr0+ ˜xj0r0. Thus,
−Hn
1(x)≥ −Hn
1(˜x) + ˜xj0r0.(21)
Further, from (19b), (10), and the fact that ˜xi0r0= 1, it follows that
−Hn
2(x) = −Hn
2(˜x) + X
r6=r0∈[k]
xi0r.(22)
Using (13), (21), (22), it follows that
Hn
c1,c2(x)≥Hn
c1,c2(˜x)−1 + c1˜xj0r0+c2X
r6=r0∈[k]
xi0r.(23)
9
In case (1), we have that ˜xj0r0= 1. Thus, from (23), we have that Hn
c1,c2(x)≥Hn
c1,c2(˜x)−1+c1>
Hn
c1,c2(˜x) = Qn
c1,c2(k, G), which contradicts the optimality of ˜xfor (20). Analogously, in case (2),
we have that Pr6=r0∈[k]˜xj0r≥1. Thus, from (23), we have that Hn
c1,c2(x)≥Hn
c1,c2(˜x)−1 + c2>
Hn
c1,c2(˜x) = Qn
c1,c2(k, G), which contradicts the optimality of ˜xfor (20).
Therefore ˜xsatisfies that there is no (i0, j 0)∈Eand r0∈[k] such that ˜xi0r0+ ˜xj0r0>1, or
i0∈[n] and r0∈[k] such that Pr∈[k]˜xi0r>1. Therefore ˜xis a feasible solution of (7), which
finishes the proof.
Next, we show that the value of the penalty parameters c1, c2in the definition of Qn
c1,c2(k, G)
in (20) can be further reduced to the values c1=c2= 1 (indeed, more generally to c1= 1, c2≥1,
or c1≥1, c2= 1), while still being able to obtain an optimal solution for the MkCS problem
for G(V, E ) by solving the QUBO problem Qn
1,1(k, G). In this case, Qn
1,1(k, G) and αk(G) are
equivalent in terms of their optimal objective value, but not necessarily in terms of their optimal
solutions. That is, the optimal solution ˜x:= arg max(Qn
1,1(k, G)) might not necessarily be a
feasible solution for the MkCS problem, which hinders the possibility of constructing a MkCS
set Hfor G(V, E ). However, as we formally show in the next corollary, the Qn
1,1(k, G) optimal
solution ˜xcan be simply modified to obtain an optimal solution for the MkCS problem.
Corollary 1 (unit-penalty nonlinear-based QUBO formulation of MkCS problem).Let k≥1
and a graph G= (V, E)on nvertices and c1, c2≥0be given, and let ˜x:= arg max{Qn
c1,c2(k, G)}
(recall (20)). If c1= 1,c2≥1or c1≥1,c2= 1, then Qn
c1,c2(k, G) = αk(G). Furthermore,
x0∈arg max{αk(G)}, where x0∈ {0,1}n×kis the output obtained when k,G(V, E ),|V|,x= ˜x
is used as input in Algorithm 1.
Proof. The result follows from the proof of Theorem 2and Algorithm 1. More specifically, in
the case c1= 1, c2≥1, notice that Algorithm 1, step (4), is equivalent to applying the mapping
Xir(·) (recall (10)) to the current solution xin the Algorithm when xir =xj r = 1 for some
(i, j)∈E,r∈[k]. Thus, it follows from (23) that the value of Hn
c1,c2=Hn
1,c2(x) can only
increase or stay equal after Algorithm 1, step (4). Similarly, in the case c1≥1, c2= 1, notice
that Algorithm 1, step (9), is equivalent to applying the mapping Xir(·) (recall (10)) to the
current solution xin the Algorithm when xir = 1, Pp6=r∈[k]xip ≥1 for some i∈[n], r∈[k].
Thus, it follows from (23) that the value Hn
c1,c2(x) = Hn
c1,1(x) can only increase of stay equal
after Algorithm 1, step (9). Thus, in both cases, at the end of Algorithm 1one obtains a feasible
solution x0for the MkCS problem with objective Hn
c1,c2(x0)≥Hn
c1,c2(˜x) = Qn
c1,c2(k, G). Since
Qn
c1,c2(k, G)≥αk(G) (see beginning of proof of Theorem 2), it follows that Qn
c1,c2(k, G) = αk(G),
and x0∈arg max{αk(G))}.
In light of Theorem 2and Corollary 1, it is natural to consider what happens if in the QUBO
problem (20) one considers penalty parameters 0 < c1, c2<1.
Proposition 1. Let k≥1and c1, c2>0be given. If c1<1or c2<1and k≥1, then there
exists a graph G(V, E)such that Qn
c1,c2(k, G)> αk(G).
Proof. First, consider the case in which 0 < c1<1, and let G(V, E) is a clique of k+ 1 vertices.
Clearly αk(G) = k. Now, for all i∈[k+ 1], r∈[k], let
xir =
1i=r, i ≤k,
1i=k+ 1, r =k,
0 otherwise.
Then, Qn
c1,c2(k, G)≥Hn
c1,c2(x) = (k+ 1) −c1> k =αk(G). Now, consider the case in which
0< c2<1, and let G(V, E) be the graph on k+ 1 vertices obtained by taking a clique in
k+ 1 vertices and adding a vertex k+ 2 and edge (k+ 1, k + 2). That is, V= [k+ 2], and
E={(i, j):1≤i<j≤k+ 1}∪{(k+ 1, k + 2)}. Clearly αk(G) = k+ 1. Now let
xir =
1i=r, i ≤k
1i=k+ 2, r =k
1i=k+ 2, r =k−1
0 otherwise
,for all i∈[k+ 2], r ∈[k].
10
Then, Qn
c1,c2(k, G)≥Hn
c1,c2(x)=(k+ 2) −c2> k + 1 = αk(G).
Remark 1. Theorem 2together with Corollary 1and Proposition 1fully characterize the QUBO
problem (20)as a means to obtain a QUBO reformulation of the MkCS problem. In short, for
any c1, c2≥1, solving the nonlinear-based QUBO problem (20)is equivalent to solving the
MkCS problem with the caveat that if either c1= 1 or c2= 1, the simple Algorithm 1might
need to be applied to the optimal solution of (20)in order to obtain an optimal solution for the
MkCS problem. On the other hand, if 0< c1<1or 0< c2<1, solving the nonlinear-based
QUBO problem (20)is not guaranteed to provide the objective value or the solution to the MkCS
problem.
As illustrated in Section 4, the full characterization provided in this section (see summary
in Remark 1) gives the freedom to fine tune the QUBO reformulation of the MkCS problem to
make the best use of quantum tools in addressing the solution of this problem.
In finishing this section, recall that the MkCS problem is equivalent to the stable set problem
when k= 1. Thus the QUBO reformulation results [see, e.g., 1,6,8,10,30,46,48,65] for the
stable set problem of the form
α(G) = max
n
X
i=1
x2
i−c1X
(i,j)∈E
xixj:x∈ {0,1}n
,(24)
for a given graph G(V, E) on nvertices and c1≥1 follow from Theorem 2and Corollary 1. In
particular, Corollary 1implies results in which c1is set to one in (24). However, Corollary 1
brings up a fact that, to the best of our knowledge, has been ignored in the literature; namely,
that when c1is set to one in (24), the support of the optimal solution of (24) might not necessarily
correspond to a stable set of the graph G(V, E). However, an optimal solution for the stable
set problem can be obtained from the optimal solution of (24) by applying Algorithm 1(see
Corollary 1).
3.3 Linear-based QUBO reformulation revisited
After the results in Section 3.2, which provide a full characterization of the QUBO problem (20)
to reformulate the MkCS problem, it is natural to consider if a similar full characterization
of the QUBO problem (12) can be obtained. Indeed, it is not difficult to see that analogous
results (with analogous proofs that are not included in the interest of brevity), to Corollary 1
and Proposition 1can be obtained for the QUBO problem (12).
Corollary 2 (unit-penalty linear-based QUBO formulation of MkCS problem).Let k≥1and
a graph G= (V, E)on nvertices and c1, c2≥0be given, and let ˜x:= arg maxx{Ql
c1,c2(k, G)}
(recall (12)). If c1= 1,c2≥1or c1≥1,c2= 1, then Ql
c1,c2(k, G) = αk(G). Furthermore,
x0∈arg max{αk(G)}, where x0∈ {0,1}n×kis the output obtained when k,G(V, E ),|V|,x= ˜x
is used as input in Algorithm 1.
Proposition 2. Let k≥1and c1, c2>0be given. If c1<1or c2<1and k≥1, then there
exists a graph G(V, E)such that Ql
c1,c2(k, G)> αk(G).
4 Benchmarking
To illustrate the benefits of the fully characterized QUBO reformulations presented in Section 3,
we next benchmark the linear-based (Section 3.1) and nonlinear-based (Section 3.2) QUBO
reformulations of the MkCS problem when solving them with a quantum annealer. For this
purpose, we present results pertaining the minimum gap [see, e.g., 52], related to the convergence
rate of (an ideal) adiabatic quantum algorithm (AQC), embedding [see, e.g., 63] into the available
quantum annealing hardware, and time-to-solution (TTS) [see, e.g., 53] when performing the
quantum annealing.
11
The embedding and TTS benchmarking results are obtained using D-Wave’s quantum an-
nealers (https://www.dwavesys.com/). Specifically, we report the different results obtained
when using two different D-Wave processors: 2000QTM and Advantage 1.1TM. The main dif-
ference between these two processors is the number of available qubits and their connectivity
within the processor. The 2000QTM processor has 2048 possible qubits (of which 2041 were
available) connected in a Chimera connectivity graph, designed as a grid of 16×16 cells of K4,4
bipartite graphs connected in a nearest-neighbor fashion by means of non-planar edges, where
each qubit is connected to at most 6 neighbors [47]. The Advantage 1.1TM processor counts
with 5640 qubits (5510 available) following a Pegasus connectivity graph, defined as three layers
of 16 ×16 cells of K4,4bipartite graphs with additional connections within and among the cells,
providing an increased connectivity for each qubit to maximum 15 neighbors [7].
Each QUBO reformulation proposed here is thus used to solve the MkCS problem in a D-
Wave quantum annealer. These numerical experiments are similar in nature to those carried
out in [10,31,61,62] to compare different QUBO formulations of various COPT problems.
To generate instances G(V, E ) for the numerical tests, given the number of nodes |V|=n,
we generate instances with randomly defined edge set E, using Erd˝os-R´enyi graphs G(n, p) with
probabilities p= 0.25,0.5,0.75 (i.e., with different levels of sparsity). In what follows, for the
purpose of brevity, we will sometimes refer to the linear-based QUBO reformulation (Section 3.1)
as L-QUBO, and to the nonlinear-based QUBO reformulation (Section 3.2) as N-QUBO. All
the classical computations are done using an Ubuntu Machine with processor Intel(R) Xeon(R)
CPU E5-2630 v4 @ 2.20GHz, 94Gb of RAM, and 20 cores.
4.1 Quantum Annealing
Before presenting the benchmarking results, we provide a very brief high level discussion about
the ideal version of the quantum annealing algorithm run by D-Wave’s quantum annealer;
that is, an AQC algorithm [following 18,19]. For additional details, the reader is directed to
[2,22,35, among many others]. The fact that complex COPT problems (such as the MkCS
problem considered here) can be reformulated as QUBO problems, means that finding the
optimal solution of the problem can be regarded as finding or sampling low-energy states from
an Ising spin model Hamiltonian Hfthat is constructed from the QUBO formulation (i.e.,
using [18, eq. (14)]). For that purpose, a simple Hamiltonian Hiwith an easily prepared ground
(low-energy) state is prepared (i.e., using [18, eq. (12)]). Then, by constructing an adequate
interpolation [see, e.g., 18, eq. (1)] between the Hiand HfHamiltonians, the system can be set
to slowly evolve (so that the adiabatic theorem [cf., 22] is satisfied) from the ground state of Hi
to a state that will yield the desired low-energy state of Hfwith high probability. In particular,
here we construct the interpolation:
H(s) = A(s)
2Hi+B(s)
2Hf,(25)
where s∈[0,1] is the adimensional (or reduced) time s=t
Twith tdenoting time and T
denoting the computation time, A(s) is the tunneling energy curve, and B(s) is the problem’s
Hamiltonian energy curve, for all s∈[0,1]. The specifications of A(s) and B(s) for both the D-
Wave 2000QTM and Advantage 1.1TM processors can be found in D-Wave’s documentation [19].
4.2 Minimum Gap
Now we consider the evolution of the AQC algorithm under (25). It is known that the minimum
gap; that is, the minimum energy gap between the lowest two energy levels during the evolution
of the AQC algorithm determines the time of the computation [see, e.g., 3,21]. Such energy
levels correspond with the eigenvalues Em(s) of the eigenstates m;siof the Hamiltonian H(s),
where m∈ {0,1,...,2n−1}in an nqubit system, and are given by [see, 18]:
H(s)|m;si=Em(s)|m;si,(26)
12
with E0(s)≤E1(s)≤ · ·· ≤ E2n−1. Thus, the minimum gap, denoted by ∆min, is given by [see,
18]:
∆min = min
s∈[0,1]{E1(s)−E0(s)}.(27)
The minimum gap ∆min provides a lower bound on the AQC computation time that is inversely
proportional to ∆2
min [see, 18, eq. (9)]; that is, the larger ∆min, the faster the AQC algorithm is
expected to converge to the ground state of the Hamiltonian Hf[see, e.g., 2].
Here, we use the exact diagonalization of the instantaneous time-dependent Hamiltonian
H(s) to compute ∆min. Although this methodology is well known to require a prohibitive
amount of computation due to the need to diagonalize matrices of size 2n×2nfor a system with
nqubits; it is suitable for the illustrative numerical tests performed here (for less computationally
expensive ways to approximately compute ∆min, we refer the readers to [3]).
In particular, to obtain the minimum gap results presented next, we begin by calculating,
for a number of finite values of s∈[0,1] (see details below), the Hamiltonian H(s) in (25) for
particular instances of the L-QUBO (resp. N-QUBO) formulation of the MkCS problem. Then,
we verify which eigenvalues of H(s) correspond to different states in the annealing process. Two
states are considered different if at the end of the annealing, their energy difference is more than
a given ε. Here, we use ε= 1 GHz. Finally, we approximately compute the minimum difference
along the annealing scaled time sof the two smallest eigenvalues corresponding to different
states (the ground state and the first excited state). The approximation comes from the fact
that the minimum in (27) is computed over a finite set of values of s∈[0,1]. Specifically, given
the monotonic behavior of A(s) and B(s), we observe that ∆min is attained at a value s∈[0,1]
that is close to the value of s∈[0,1] in which the maximum of the minimum eigenvalue
is attained. Moreover, for the instances considered here, the minimum eigenvalue is concave
on s∈[0,1], allowing us to more efficiently sample the domain [0,1] in search for the value
of ∆min. Namely, we first consider a coarse discretization of the domain [0,1] (taking only 10
equally spaced elements of the set) and then determine the three points whose middle point
would be larger than both its neighbors. This interval contains the values of s∈[0,1] in which
both the maximum value of the smallest eigenvalue, and ∆min is attained. We sample this
interval by computing 10 points between each of these points (including them) to refine the
approximation to the value s∗in which the minimization in (27) is attained. This procedure
only requires 44 computations of the eigenvalues of the Hamiltonian H(s).
4.2.1 Minimum Gap Results
Next, we compare the minimum gap (∆min) resulting when using the L-QUBO reformulation
and the N-QUBO reformulations for the MkCS problem with varying penalty parameters (i.e.,
c1,c2) on small instances of the MkCS problem.
In particular, Figures 1and 2compare the ∆min obtained from the L-QUBO (12) and N-
QUBO (20) for instances of the MkCS problem in which k= 1, where the underlying graphs are
randomly selected G(5,0.25) and G(5,0.75) graphs. These bar plots, as well as the remaining
ones in this section, provide information about the distribution of ∆min. Specifically, each bar is
obtained by computing ∆min on 100 randomly generated graphs G(5, p), p∈ {0.25,0.50,0.75},
for different values of the penalization parameters. The tick line, represents the median of ∆min,
the black diamond represents ∆min, the average of ∆min, the bar encompasses the values within
the 25% and 75% quantiles of ∆min’s distribution, and the dotted interval encompasses the
values within the 0% and 100% of ∆min’s distribution.
From Figures 1and 2, it follows that the N-QUBO results in higher ∆min than the L-
QUBO; therefore, in theory, the N-QUBO Hamiltonian should converge faster to a low energy
state than the L-QUBO Hamiltonian. We can state this more formally by performing a simple
hypothesis test. Let ∆a
min(k, p, c1, c2) (resp. µ∆a
min (k, p, c1, c2)) be the average (resp. mean) of
the minimum gap for the a-QUBO formulation of the MkCS instance from G(5, p) graphs, and
13
Figure 1: k= 1, G(5,0.25). Figure 2: k= 1, G(5,0.75).
penalty parameters c1,c2. Then, consider the hypothesis test:
Ho:µ∆L
min (1, p, 1,·)≥µ∆N
min (1, p, 1,·)−δ∆N
min(1, p, 1,·)
Ha:µ∆L
min (1, p, 1,·)< µ∆N
min (1, p, 1,·)−δ∆N
min(1, p, 1,·),(28)
where δ∈[0,100%]. That is, in (28) we are statistically comparing the left-most bars of the
N-QUBO subplot and the L-QUBO subplot of Figures 1and 2(for brevity, the results for the
case p= 0.50 have not bee plotted), under the null hypothesis that the L-QUBO provides a
higher mean ∆min. Then, for any p∈ {0.25,0.50,0.75}one gets that the null hypothesis Ho
in (28) can be rejected with 95% confidence for values of δup to 2%. Thus, loosely speaking,
the N-QUBO results in values of ∆min that on average are 2% higher than the ones obtained
by the L-QUBO, when using penalty parameter c1= 1.
One advantage of having the full characterization of the penalty constants, for which the
QUBO formulations (12) and (20) become reformulations of the MkCS problem, is that we can
investigate what are the trade-offs of increasing such penalty values from their minimum ones.
Intuitively, one might expect that ∆min increases (faster convergence) as the values of the penalty
parameters c1, c2increase. This reasoning stem from the fact that higher penalty parameters
increase the suboptimality of infeasible solutions of the original problem in its associated QUBO
reformulation. However, from Figures 1(left) and 2(left) it follows that ∆min remains fairly
unchanged as the penalty parameter c1increases. More formally consider a similar hypothesis
test to the one considered in (28).
Ho:µ∆N
min (1, p, c1,·)≥µ∆N
min (1, p, 1,·) + δ∆N
min(1, p, 1,·)
Ha:µ∆N
min (1, p, c1,·)< µ∆N
min (1, p, 1,·) + δ∆N
min(1, p, 1,·).(29)
Then, for any p∈ {0.25,0.50,0.75)}, c1∈ {2,5}, one gets that the null hypothesis Hoin (28)
can be rejected with 95% confidence for values of δless than 1%. Thus, loosely speaking, the
N-QUBO with penalty parameter c1= 1 results in values of ∆min that on average are not 1%
lower than the ones obtained by the N-QUBO with higher penalty parameters c1∈ {2,5}.
It is clearly interesting to investigate how the characteristics above look when considering
higher values of k. Due to the complexity of the exact diagonalization procedure used to compute
∆min, we limit to the study of the case k= 2 for the N-QUBO reformulation, considering penalty
parameters constants c1∈ {1,2,5}, and c2∈ {1,2,5}.
Much like in the case when k= 1, from Figures 3and 4it follows that ∆min barely increases
as the penalty parameters c1, c2increase, in comparison to the value of ∆min when the penalty
parameters are set to c1=c2= 1. Statistically, things are a bit different. Formally, consider a
14
Figure 3: k= 2, G(5,0.25). Figure 4: k= 2, G(5,0.75).
similar hypothesis test to the one considered in (29).
Ho:µ∆N
min (2, p, c1, c2)≥µ∆N
min (2, p, 1,1) + δ∆N
min(2, p, 1,1)
Ha:µ∆N
min (2, p, c1, c2)< µ∆N
min (2, p, 1,1) + δ∆N
min(2, p, 1,1).(30)
Table 1shows the minimum value of δin (30) for which the null hypothesis Hoin (29) can be
rejected with a 95% confidence level. Loosely speaking, the value of δindicates the percentage
by which the value of ∆min(2, p, 1,1) must be higher in order to reject the hypothesis that larger
penalty parameters (i.e., larger than c1=c2= 1) result in a larger mean value of ∆min.
δ
(c1, c2)p= 0.25 p= 0.50 p= 0.75 Ho
(1,2) -5% 8% 12% Reject
(1,5) -5% 5% 11% Reject
(2,1) 1% 4% 1% Reject
(2,2) 1% 12% 7% Reject
(2,5) 1% 13% 6% Reject
(5,1) 1% 2% 0% Reject
(5,2) 4% 15% 11% Reject
(5,5) 11% 22% 13% Reject
Table 1: Hypothesis test (30) for different parameters with 95% confidence.
Overall, there is a recognizable pattern in Table 1. Namely, it is clear that the sparser the
underlying graph (i.e., lower probability p) used to construct the instance of the MkCS problem,
the smaller the effect of increasing penalty parameters is on increasing the mean of ∆min (i.e.,
accelerating convergence of an AQC algorithm). This is intuitively expected, given that sparsity
in the underlying graph results in lower number of penalty terms in the N-QUBO (20). In
particular, notice that a significant increase in the mean of ∆min, for sparse underlying graphs
(i.e., p= 0.25), only arises when the penalty parameters are increased from c1=c2= 1 to
c1=c2= 5. In contrast, for non-sparse underlying graphs (i.e., p= 0.75), increases in the
penalty constants above c1=c2= 1 bring increases in the mean of ∆min of about 10% in
most cases. In Section 4.4, we will analyze how these increases in the mean of ∆min affect the
convergence to a solution in D-Wave’s quantum annealing devices. Before doing this analysis,
we first consider the differences in terms of embedding requirements between the N-QUBO and
L-QUBO formulation.
15
4.3 Embedding
Current NISQ devices have a low number of qubits available with restricted connectivity. Given
this, the number of qubits required to embed [see, e.g., 63] a QUBO reformulation of a given
COPT problem in a quantum device is a very important benchmark to compare the benefits of
different QUBO reformulations of a COPT problem [see, e.g., 25,31,61,62]. Next, to benchmark
the N-QUBO (20) versus the L-QUBO (12) reformulation of the MkCS problem, we use the
number of qubits needed to embed the QUBO reformulation in both a 2048 qubits Chimera
connectivity graph (for D-Wave’s 2000QTM processor), and a 5640 qubits Pegasus connectivity
graph (for D-Wave’s Advantage 1.1TM processor). For this purpose, we use D-Wave’s embedding
algorithm [see, e.g., 5,68, for a discussion of different embedding algorithms].
Figure 5: Embeding: k= 1, G(n, 0.25). Figure 6: Embedding: k= 1, G(n, 0.25).
Note that for a graph G(V, E ) with nnodes, the number of binary variables required to
formulate the L-QUBO for the associated MkCS problem is k(n+|E|+ 1), while kn binary
variables are needed to formulate the associated N-QUBO. Not surprisingly, the N-QUBO would
require less number of qubits than the L-QUBO when these QUBOs are embedded into D-Wave’s
quantum annealers. The following results show how the increased number of binary variables
required by the L-QUBO affects the difference between the qubits required to embed both
QUBO formulations.
Figure 7: Embeding: k= 1, G(n, 0.75). Figure 8: Embedding: k= 1, G(n, 0.75).
16
In Figures 5-12, the number of average qubits required by both the L-QUBO and the N-
QUBO formulations are plotted for values of k∈ {1,2,5}, graphs G(n, p) for values of n∈[5,50],
p∈ {0.25,0.50,0.75}, and D-Wave’s 2000QTM and Advantage 1.1TM processors. The average
is computed over five (5) random graphs G(n, p) generated for each combination of n, p values,
as well as ten (10) runs of D-Wave’s embedding algorithm. The bars plotted with each point in
the graph represent the values within one standard deviation of the average value.
Figure 9: Embeding: k= 2, G(n, 0.50). Figure 10: Embedding: k= 2, G(n, 0.50).
From all Figures 5-12, it is clear that in terms of embedding requirements, the N-QUBO
formulation is substantially better than the L-QUBO formulation. This is true not only in
terms of the average qubits required to embed each QUBO, but the volatility of the number
of qubits required to embed each QUBO. In Figure 5, in which sparse graphs (i.e., p= 0.25)
are used for the case k= 1, both QUBO formulations can be embedded, for graphs with up
to n= 50, in D-Wave’s 2000QTM processor. However, in Figure 7, where dense graphs (i.e.,
p= 0.75) are considered, now the L-QUBO can be embedded only for graphs with up to n= 40.
From Figures 9and 11 it is clear that as kand pincrease, this trend of being able to embed
larger problems in terms of number of nodes ncontinues to be evidenced even more. Even using
the more powerful Advantage 1.1TM processor, Figure 12 shows that for sparse graphs (i.e.,
p= 0.25) and k= 5, the L-QUBO can only be embedded for graphs with up to n= 40, while
it seems that the N-QUBO can be embedded for graphs with up to n= 80 (i.e., the double
number of nodes).
By pairwise comparing Figures 5,7,9,11, versus Figures 6,8,10,12, one can see the
advantages of the Advantage 1.1TM processor versus the 2000QTM processor. The effect of
having a larger number of qubits clearly means that larger instances of the L-QUBO can be
embedded in the Advantage 1.1TM processor than in the 2000QTM processor. Also evident
are the effects of the improved connectivity between qubits in the Advantage 1.1TM processor.
Namely, it is clear that the Advantage 1.1TM processor is able to embed QUBO problems using
a substantially lower number of qubits than in the 2000QTM processor. For example, from
Figures 7and 8, it takes about 800 qubits to embed the N-QUBO of graphs with 50 nodes in
the 2000QTM processor, while it takes about half the number of qubits (about 400) to embed the
N-QUBO of graphs with 50 nodes in the Advantage 1.1TM processor. The pairwise comparison
between Figures 5,7,9,11, and Figures 6,8,10,12, also shows how the added connectivity in
the Advantage 1.1TM processor clearly lowers the volatility of the number of qubits required to
embed a QUBO using D-Wave’s embedding algorithm.
17
Figure 11: Embeding: k= 5, G(n, 0.25). Figure 12: Embedding: k= 5, G(n, 0.25).
4.4 Time-To-Solution
We now finish our numerical tests by comparing (mirroring some of the tests in Sections 4.2.1
and 4.3) the time-to-solution (TTS) required, by both D-Wave’s quantum annealer processors,
when using the N-QUBO and L-QUBO reformulation of the MkCS, with penalty parameters
c1∈ {1,2,5}, ce∈ {1,2,5}, for values k= 2, on random graphs G(n, p) for n∈[5,50], p∈
{0.25,0.75}(similar results were obtained for k∈ {1,5}, and p= 0.50 but are not presented for
brevity). Besides benchmarking the N-QUBO versus the L-QUBO reformulation of the MkCS,
these tests will be used to analyze the effect of the value of penalization constants in the TTS,
and the effect in the TTS of using the more powerful Advantage 1.1TM processor.
Figure 13: k= 2, G(n, 0.25), c1=c2= 1. Figure 14: k= 2, G(n, 0.25), c5=c2= 5.
TTS is a common benchmark used to evaluate the performance of quantum annealers, and
is defined as the expected time required to find a ground state of the desired Hamiltonian with
a level of confidence α, which is set to 95% in our tests. Formally [see, e.g., 53],
TTS = trun
ln(1 −α)
ln(1 −p),(31)
where trun, fixed to 20µs in our tests, is the running time elapsed in a single run of the quantum
annealer, and pis the probability of finding the ground state of the desired Hamiltonian. In our
18
tests, pis estimated by running the quantum annealer 1000 times.
Figure 15: k= 2, G(n, 0.25), c1=c2= 1. Figure 16: k= 2, G(n, 0.25), c5=c2= 5.
From Figures 13-20, it is clear that regardless of the quantum annealing processor, penalty
parameters, or sparsity of the graph, the N-QUBO reformulation of the MkCS problem performs
substantially better than the associated L-QUBO reformulation in terms of TTS. For example,
note that from Figure 15 it follows that for sparse graphs (i.e., p= 0.25), the N-QUBO results
in TTS values that are between three (3) orders of magnitude faster, for small graphs, and one
(1) order of magnitude faster, for larger graphs, than the associate TTS values for the L-QUBO.
Figure 17: k= 2, G(n, 0.75), c1=c2= 1. Figure 18: k= 2, G(n, 0.75), c5=c2= 5.
From Figure 17 it is clear that when considering non-sparse graphs (i.e., p= 0.75) in the
2000QTM processor, the advantages of the N-QUBO over the L-QUBO in terms of TTS only
increase. In particular, notice that while a trun of 20µs is enough to find the optimal solution
with some small probability for instances of the MkCS problem with underlying graphs of up
to n= 50 nodes. In contrast, once the number of nodes of the underlying graph goes beyond
n= 25, with the L-QUBO the quantum annealer is unable to find any optimal solution in all
1000 runs of 20µs.
Not surprisingly, pairwise comparing Figures 13,17, with Figures 15,19, the advantages of
the Advantage 1.1TM processor over the 2000QTM processor in terms of TTS are clear. For the
L-QUBO the newer processor finds solutions for much larger instances of the MkCS problem.
19
Figure 19: k= 2, G(n, 0.75), c1=c2= 1. Figure 20: k= 2, G(n, 0.75), c5=c2= 5.
For the N-QUBO, the rate of increase of the TTS as the size of the MkCS problem increases is
about one order of magnitude lower in the newer processor.
We can also use the results presented in this question to study whether the conclusions made
in Section 4.2.1, particularly in Table 1, reflect on the actual TTS time when using a quantum
annealer. Note that from Table 1it was expected that for instances of the MkCS problem with
k= 2 and underlying graphs G(n, 0.25), increasing the penalty constants from c1=c2= 1 to
c1=c2= 5 would result in a faster convergence. However, by comparing Figures 13 and 14, as
well as Figures 15 and 16, it follows that increasing the penalty constants in this way is actually
counterproductive for both quantum annealing processors in terms of TTS (i.e., in Figures 14
and 16, the “slope” at which the TTS increases with the number of nodes is higher. Also, from
Table 1it was expected that for instances of the MkCS problem with k= 2 and underlying
graphs G(n, 0.75), the increase in the penalty constants from c1=c2= 1 to c1=c2= 5 would
have an even slightly higher benefit in terms of speed of convergence (compared with G(n, 0.25)
graphs). However, by comparing Figures 17 and 18, as well as Figures 19 and 20, it follows
that increasing the penalty constants in this way does not produce discernible improvements for
the quantum annealing processors in terms of TTS. Most likely, this means that any theoretical
advantages in terms of convergence obtained by increasing the value of penalty constants is
off-set by the precision problems that using larger penalty parameters brings for the quantum
annealing processors in practice. The fact that being able to use penalty parameters close to
one (1) is beneficial for quantum annealers is discussed, for example, in [63]. This shows the
importance of being able to fully characterize the range of penalty constants that result in a
QUBO being a reformulation of a COPT problem.
5 Concluding remarks
In this paper, we consider a particularly important combinatorial optimization (COPT) prob-
lem; namely, the maximum k-colorable subgraph (MkCS) problem, in which the aim is to find an
induced k-colorable subgraph with maximum cardinality in a given graph. This problem arises
in channel assignment in spectrum sharing networks (e.g., Wi-Fi or cellular), VLSI design,
human genetic research, cybersecurity, cryptography, and scheduling. We derive two QUBO
reformulations for the MkCS problem; a linear-based QUBO reformulation (Theorem 1) and
a nonlinear-based QUBO reformulation (Theorem 2). Furthermore, we fully characterize the
range of the penalty parameters that can be used in the QUBO reformulation. In the case of the
linear-based QUBO reformulation, this analysis shows that Theorem 1provides a better QUBO
reformulation for the MkCS problem than the one that could be obtained using the QUBO re-
20
formulation techniques recently introduced by Lasserre [39]. In the case of the nonlinear-based
QUBO reformulation, this analysis shows that Theorem 2provides a better QUBO reformula-
tion for the MkCS problem than the one that could be obtained using the well-known QUBO
reformulation techniques introduced by Lucas [42]. Our proofs bring forward a fact that is over-
looked in related articles. Namely, that when minimal penalty parameters are used in QUBO
reformulations, the equivalence in terms of objective value between a problem and its associ-
ated QUBO reformulation does not necessarily mean that the optimal solution of the QUBO
reformulation provides a feasible, optimal solution for the original problem. This is shown to be
the case for the MkCS problem in Corollaries 1and 2. Given that for k= 1 the MkCS problem
is equivalent to the stable set problem, we show (see end of Section 3.2) that this issue applies
to the well-known QUBO reformulation of the stable set problem (24). However, we show that
this issue can be be simply addressed by using the greedy Algorithm 1(for general instances of
the MkCS problem).
We finish in Section 4by illustrating the advantages of the nonlinear-based QUBO reformula-
tion over the linear-based QUBO reformulation in terms of embedding requirements, convergence
rate, and time-to-solution when the QUBO reformulations are used to solve the MkCS problem
in a quantum annealing device. The experiments also illustrate the importance of having a full
characterization of the penalty parameters that ensure the proposed QUBOs are indeed reformu-
lations of the original problem. For example, we explore the potential theoretical and practical
gains of using higher penalty parameters than the minimum ones required for the QUBO to
become a reformulation of the MkCS problem. Our results show that although there are some
theoretical benefits of using larger than minimal penalty parameters, they do not translate to a
faster convergence to a solution of the problem on a quantum annealing computing device.
Our results contribute to recent literature that beyond obtaining QUBO reformulations of
COPT problems such as the graph isomorphism problem as well as tree and cycle elimination
problems, look for improved QUBO reformulations of these problems for NISQ devices [see, e.g.,
10,25,31,61,62]. That is, QUBO reformulations that are tailored to be more efficiently used
in NISQ devices.
Acknowledgements
This project has been carried out thanks to funding by the Defense Advanced Research Projects
Agency (DARPA), ONISQ grant W911NF2010022, titled The Quantum Computing Revolution
and Optimization: Challenges and Opportunities. The project was also supported by the Oak
Ridge National Laboratory OLCF grant ENG121, which provided the authors with in-kind
access to D-Wave’s quantum annealers. The second author acknowledges the support of the
Center for Advanced Process Decision Making (CAPD) at Carnegie Mellon University.
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