A heuristic for max-cut in toroidal grid graphs

Abstract

The paper reports on some preliminary results obtained by using polynomial time algorithms to solve max-cut on some special graphs within a heuristic scheme known as subgraph sampling. Computational results are reported on toroidal graphs of about 100 000 nodes with edge weights generated from both a Gaussian and a bivariate uniform distribution.

Full text

A heuristic for max-cut in toroidal grid graphs
Claudio Gentile1, Giovanni Rinaldi1, Esteban Salgado1,2, Bao Duy Tran3
1Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti” – CNR, Italy
<name>.<surname>@iasi.cnr.it
2Department of Computer, Control and Management Engineering Antonio Ruberti, Sapienza
University of Rome, Italy
3Ruprecht-Karls-University Heidelberg, Germany
baoduy.tran@uni-heidelberg.de
Abstract
The paper reports on some preliminary results obtained by using polynomial time
algorithms to solve max-cut on some special graphs within a heuristic scheme known
as subgraph sampling. Computational results are reported on toroidal graphs of about
100 000 nodes with edge weights generated from both a Gaussian and a bivariate uniform
distribution.
Keywords :max-cut, toroidal grid graphs, Ising spin glass, heuristics, subgraph sampling.
1 Introduction
One of the most famous applications of the max-cut problem is the computation of the ground
state of a spin glass under the Ising model, a well-studied issue in Statistical Physics.
Such a computation amounts to solving a quadratic unconstrained binary optimization or,
equivalently, a max-cut problem, on a 2-dimensional (L×M)or on a 3-dimensional (L×M×N)
grid graph, whose size is typically very large in order to simulate a real physical system.
The edge weights of such a graph are randomly generated either from a zero-mean Gaussian
distribution or from a bivariate uniform distribution with values in the set {−1,1}. Two are
the requirements that the instances under consideration have to satisfy: a) Mand Nmust be
sufficiently large; b) to avoid the border effects that would deviate the behavior of the model
from the one of a real system, the grid is “wrapped around” along each of the three dimensions.
Thus, in 2D, for example, the graph to consider becomes a toroidal grid.
In the Physics community, these problems are solved with Monte Carlo techniques, which
have two drawbacks: they tend to be slow to converge to an accurate solution when the graph
sizes are large and do not provide any measure of how much the solution deviates from the
true optimum.
Max-cut is solvable in polynomial time on planar graphs and 2D grids or 2D grids wrapped
along only one direction are of this kind. Exact solutions for grid instances of sizes up to
3000 ×3000 are reported in [7]. The problem has been proven to be polynomially solvable on
toroidal grids with bounded weights in [3], but the algorithm that achieves this result is not
very practical for large instances. For example, solutions of problems up to only 50 ×50 are
reported in [8]. Branch-and-cut methods have been proposed to solve the problem on toroidal
grids (see, e.g., [6] for a survey). The largest instances reported in these studies are 150×150 for
the Gaussian weights and 100 ×100 for the ±1ones. Therefore, there is a demand for fast and
accurate heuristic methods that would make it possible to run simulations on a large sample of
large instances and would improve the performances of the exact methods that usually benefit
from the knowledge of a feasible solution of good quality.
In the context of the Markov Random Field problem, a heuristic was proposed in [4]. The
idea of this heuristic, which is now referred to as the subgraph sampling method, was applied by

Selby in [11] to max-cut for a class of graphs known as the Chimera graphs. Selby’s heuristic
was extensively applied to solve max-cut on a large variety of instances of Chimera graphs
in [5] and was shown to be very fast and able to find the optimum for all the instances of
the test-bed. The central part of Selby’s method relies on the ability to solve the problem on
subgraphs of low tree-width efficiently. The motivation of this paper is to use the subgraph
sampling method with a different type of subgraphs for which max-cut is polynomially solvable
and to evaluate the efficiency of this new approach on toroidal grid instances with Gaussian
and ±1edge weights.
2 Algorithms
Given a weighted graph G= (V, E, w), the max-cut problem calls for a partition (W:V\W)
of the node-set that defines an edge-cut of maximal weight. The two sets of the partition are
called the shores of the cut. We represent the cut defined by (W:V\W)with a vector
x∈ {−1,1}V, where for v∈V,xv= 1 if u∈Wand xv=−1otherwise. Then the weight of
such a cut is given by
X
ij∈E
wij (1 −xixj)/2.
A variable transformation of the kind x0
v=−xv, which is called switching on node vin
the max-cut literature, transforms a max-cut instance into an equivalent one and will be very
useful in the following because it serves a dual purpose:
a) suppose that at a given step of an algorithm we want to impose that all nodes of a subset
U⊂Vbelong to the same shore. We can apply the switching operation to some nodes
of Uso that all its nodes get the same x-value. At this point, we contract all nodes of
Uinto a single node. The resulting graph, that we denote by GU, has the property that
all its cuts correspond to all cuts of Gwhere all nodes of Ubelong to the same shore.
b) suppose that we need to flip the weight sign of the edges incident with a node v. Then
switching on vachieves what we want.
We now outline the subgraph sampling scheme and the two heuristics that used it.
Subgraph sampling scheme.
(i) At the beginning of the algorithm, a random assignment of the nodes to the two shores
is applied, i.e., a point ˆx∈ {1,−1}Vis randomly chosen.
(ii) A subset U⊂Vis “suitably” chosen. We then find the best among all cuts where the
nodes in V\Uare constrained to keep the value given in ˆxand replace ˆxwith the value
corresponding to such a cut. This is achieved by possibly applying the switching to some
nodes of V\Uand contracting the nodes of V\Uto generate the graph GV\U. In order
for the scheme to be of practical use, max-cut for the instance GV\Uhas to be efficiently,
or — even better — polynomially solvable.
(iii) Until a maximum number of non-improving iterations is not reached, select a new node-
set U⊂Vand iterate step (ii).
(iv) Perturb the current vector ˆxand repeat from step (ii) until no more improvements take
place, despite perturbation.
(v) If additional computing time is allowed, the node assignment is randomly generated
afresh and the process is restarted from step (ii). This way, the Borel-Cantelli lemma
assures that the optimal solution is theoretically eventually attained.
(vi) Finally, return the best solution found during the whole process.

Even if this scheme is general, here we will apply it to 2-dimensional (L×M)grid graphs
generating two algorithms depending on the way the set Uis chosen.
Planar subgraph. Given a row r, with node-set R, and a column c, with node-set C, we
define U=V\ {R, C}. The successive selections of the set Uare performed by “moving” the
column and the row, i.e., U0=V\{r0, c0}where r0= (r+1)( mod L)and c0= (c+1)( mod M).
In this case, the resulting graph GUis planar (a grid with an external node connected only
to the boundary of the grid); therefore, max-cut can be solved in polynomial time on the
respective GUgraph by finding a perfect matching in an auxiliary graph constructed in the
same way as described in [7]. We observed that with sizes up to 100 ×100 the instances can
be solved optimally using the algorithm described in [6] and very often planar subgraph solves
them optimally, for example, for the bivariate case.
Negative subgraph. Let Ube a non-empty node-set that induces an acyclic subgraph of
G(Ucould even be made of a single node); then we define a negative subgraph Fby the
following procedure: Repeat until no more changes apply: pick a node in v∈V\Usuch that
N(v)∩U6=∅1. If the weights of all edges (v, u)∈Efor u∈Uhave the same sign, add vto U.
It is easy to see that, by applying switching, we can make all edges of Fto have a non-positive
weight. Therefore, in GUall edges, except those that are incident with the node obtained by
the contraction of V\U, have non-positive weight and max-cut can be solved in polynomial
time, as shown in [9], using max-flow techniques.
The set U=U(t, p)with t∈ {1, . . . , L}and p∈ {0,1}from which we start the construction
of Fis defined by the nodes of row tand column (p+j)(mod M)for j∈ {1, . . . , M −1}, by
the nodes of row (t+i)(mod L)for i∈ {1, . . . , L −2}and column (p+j)for all even j∈
{1, . . . , M −1}, and by the node defined by row (t−1)(mod L)and column (p−1)(mod M).
The successive selections of the set Uare performed by “moving” the parameters to U(t0, p0)
where t0= (t+ 1)( mod L)and p0= 1 −p. Notice that the graph induced by U(t, p)is a forest,
thus it is acyclic. Likewise, interchanging rows with columns, another sequence of U(t, p)sets
is produced. Note that if U(t, p)is defined as above, its complement also induces an acyclic
subgraph. Therefore, the scheme can also be modified to solve max-cut exactly on both the
graph generated by U(t, p)and the one generated by its complement and then move to the
next U(t0, p0). With this modification, each step of the scheme is applied to the full graph
solving two max-cut problems. We also notice that the sets generated by U(t, p)and by its
complement typically overlap, thus we are optimizing on overlapping subsets of V.
3 Numerical results
To measure the quality of these heuristics we tested them on 316 ×316 toroidal grids (with
approximately 100 000 nodes), see Tab. 1. Tests were performed on a machine running Ubuntu
18.04.4 LTS with an Intel Xeon Gold 6136 Processor with 3.0 Ghz base frequency. The
implementations described in [2] and in [1] were used for the matching and the max-flow
algorithms, respectively. Graphs were randomly generated using rudy [10]. In the table header
pindicates the proportion of positive/negative edges in a graph with ±1edge-weights, sdenotes
the seed and Gdenotes Gaussian weights. We use the value (val) obtained using planar
subgraph (P) and compare it with the results of negative subgraph with and without the use
of the complement. For each graph and method, we ran the algorithm 10 times and show the
average (avg) number of iterations needed to obtain the best solution (it), the average (avg)
value (val), the average (avg) and standard deviation (std) of the percentage relative difference
with the planar subgraph (%difP) and the average (avg) time needed to obtain the optimal
solution. For negative subgraph the maximum time was set to 60 minutes (the maximum time
condition is verified at the beginning of each iteration).
1By N(v)we denote the nodes of Vadjacent to v.

P No complement Complement
p s val time it (avg) val (avg) %difP (avg/std) time (avg) it (avg) val (avg) %difP (avg/std) time (avg)
40
1 89968 1612.79 6.70 89770.40 0.22 / 0.02 3681.97 3.20 89772.60 0.21 / 0.02 4013.31
2 89978 1594.09 6.40 89768.80 0.23 / 0.01 3651.78 3.90 89779.00 0.22 / 0.01 4235.11
3 90062 2438.43 5.70 89848.00 0.24 / 0.02 3499.65 2.80 89847.60 0.24 / 0.02 3857.10
4 89952 1553.53 6.40 89731.60 0.25 / 0.01 3822.29 3.50 89735.80 0.24 / 0.02 3946.81
50
1 69984 1626.84 6.30 69773.20 0.30 / 0.02 3504.64 2.90 69780.00 0.29 / 0.01 3912.82
2 70002 2444.74 6.50 69794.00 0.29 / 0.01 3701.30 3.30 69798.40 0.29 / 0.02 3977.30
3 69956 1480.32 6.60 69741.60 0.31 / 0.02 3632.59 3.20 69749.00 0.30 / 0.02 3917.03
4 70064 1473.25 6.50 69850.20 0.31 / 0.01 3812.63 3.30 69855.80 0.30 / 0.01 4048.55
60
1 49920 1609.70 7.40 49715.80 0.41 / 0.02 3717.88 3.60 49714.40 0.41 / 0.02 3850.08
2 50094 1926.58 6.40 49879.20 0.42 / 0.02 3562.71 3.20 49888.80 0.40 / 0.02 3869.57
3 50074 2485.99 6.70 49876.40 0.39 / 0.02 3662.16 2.90 49881.00 0.39 / 0.02 4035.61
4 50000 1504.92 6.50 49791.40 0.42 / 0.01 3678.25 3.40 49804.80 0.39 / 0.02 4119.61
G
1 6528526722 1673.16 5.20 6457847194.60 1.08 / 0.02 3855.91 3.10 6452574600.70 1.16 / 0.03 4104.11
2 6554507128 1688.89 5.50 6484463753.50 1.06 / 0.02 3871.09 3.30 6480230701.70 1.13 / 0.03 4067.86
3 6548455479 1706.82 5.10 6476848337.70 1.09 / 0.03 3938.77 3.10 6473079819.30 1.15 / 0.03 4353.24
4 6545055776 1659.93 5.20 6474477296.90 1.08 / 0.03 3870.45 3.50 6471527989.50 1.12 / 0.05 4297.28
TAB. 1: Results with and without complement for 316 ×316 toroidal grids.
As we can observe negative subgraph presents solutions that are comparable within a relative
difference between 0.21 and 0.42 for graphs with edge-weights in {−1,1}. When the weights
are normally distributed errors rise up to 1.08. We also observe that the complement approach
improves the solution on the {−1,1}case, which is not true with the Gaussian weights. A
major difference between both approaches is that negative subgraph can be used for any graph.
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