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Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions

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We study a new optimization problem that minimizes the ratio of two monotone k-submodular functions. The problem has applications in sensor placement, influence maximization, and feature selection among many others where one wishes to make a tradeoff between two objectives, measured as a ratio of two functions (e.g., solution cost vs. quality). We develop three greedy based algorithms for the problem, with approximation ratios that depend on the curvatures and/or the values of the functions. We apply our algorithms to a sensor placement problem where one aims to install k types of sensors, while minimizing the ratio between cost and uncertainty of sensor measurements, as well as to an influence maximization problem where one seeks to advertise k products to minimize the ratio between advertisement cost and expected number of influenced users. Our experimental results demonstrate the effectiveness of minimizing the respective ratios and the runtime efficiency of our algorithms. Finally, we discuss various extensions of our problems.
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Algorithms for Optimizing the Ratio of
Monotone k-Submodular Functions
Hau Chan1( ), Grigorios Loukides2, and Zhenghui Su1
1University of Nebraska-Lincoln, Nebraska, USA
hchan3@unl.edu, zsu@huskers.unl.edu
2King’s College London, London, UK
grigorios.loukides]@kcl.ac.uk
Abstract. We study a new optimization problem that minimizes the
ratio of two monotone k-submodular functions. The problem has applica-
tions in sensor placement, influence maximization, and feature selection
among many others where one wishes to make a tradeoff between two
objectives, measured as a ratio of two functions (e.g., solution cost vs.
quality). We develop three greedy based algorithms for the problem, with
approximation ratios that depend on the curvatures and/or the values of
the functions. We apply our algorithms to a sensor placement problem
where one aims to install ktypes of sensors, while minimizing the ratio
between cost and uncertainty of sensor measurements, as well as to an
influence maximization problem where one seeks to advertise kproducts
to minimize the ratio between advertisement cost and expected number
of influenced users. Our experimental results demonstrate the effective-
ness of minimizing the respective ratios and the runtime efficiency of our
algorithms. Finally, we discuss various extensions of our problems.
Keywords: k-submodular function ·greedy algorithm ·approximation
1 Introduction
In many applications ranging from machine learning such as feature selection
and clustering [1] to social network analysis [14], we want to select kdisjoint
subsets of elements from a ground set that optimize a k-submodular function.
Ak-submodular function takes in kdisjoint subsets as argument and has a
diminishing returns property with respect to each subset when fixing the other
k1 subsets [6]. For example, in sensor domain with ktypes of sensors, a k-
submodular function can model the diminishing cost of obtaining an extra sensor
of a type when fixing the numbers of sensors of other types. Most recently, the
problem of k-submodular function maximization has been studied in [6, 7,14, 18].
In this work, we study a new optimization problem which aims to find k
disjoint subsets of a ground set that minimize the ratio of two non-negative and
monotone k-submodular functions. We call this the RS-kminimization problem.
The problem can be used to model a situation where one needs to make a trade-
off between two different objectives (e.g., solution cost and quality). For the
exposition of our problem, we provide some preliminary definitions.
2 Hau Chan et al.
Let V={u1, ..., un}be a non-empty set of nelements, k1 be an inte-
ger, and [k] = {1, . . . , k}. Let also (k+ 1)V={(X1, ..., Xk)|XiV, i
[k],and XiXj=,i6=j[k]}be the set of k(pairwise) disjoint subsets
of V. A function f: (k+ 1)VRis k-submodular [6] if and only if for any
x,y(k+ 1)V, it holds that:
f(x) + f(y)f(xuy) + f(xty),(1)
where
xuy= (X1Y1, ..., XkYk) and
xty= ((X1Y1)\([
i[k]\{1}
XiYi), ..., (XkYk)\([
i[k]\{k}
XiYi)).
When k= 1, this definition coincides with the standard definition of a submod-
ular function. Given any x= (X1, . . . , Xk)(k+ 1)Vand y= (Y1, . . . , Yk)
(k+ 1)V, we write xyif and only if XiYi,i[k]. The function
f: (k+ 1)VRis monotone if and only if f(x)f(y) for any xy.
Our RS-kminimization problem aims to find a subset of (k+ 1)Vthat mini-
mizes the ratio of non-negative and monotone k-submodular functions fand g:
min06=x(k+1)Vf(x)
g(x). The maximization version can be defined analogously.
Applications. We outline some specific applications of our problem below:
1. Sensor placement: Consider a set of locations Vand kdifferent types of sen-
sors. Each sensor can be installed in a single location and has a different purpose
(e.g., monitoring humidity or temperature). Installing a sensor of type i[k] in-
curs a certain cost that diminishes when we install more sensors of that type. The
cost diminishes, for example, because one can purchase sensors of the same type
in bulks or reuse equipment for installing the sensors of that type for multiple
sensor installations [8]. We want to select a vector x= (X1, . . . , Xk) containing
ksubsets of locations, each corresponding to a different type of sensors, so that
the sensors have measurements of low uncertainty and also have small cost. The
uncertainty of the sensors in the selected locations is captured by the entropy
function H(x) (large H(x) implies low uncertainty) and their cost is captured
by the cost function C(x); H(x) and C(x) are monotone k-submodular [8, 14].
The problem is to select xthat minimizes the ratio C(x)
H(x).
2. Influence maximization: Consider a set of users (seeds)Vwho receive in-
centives (e.g., free products) from a company, to influence other users to use
kproducts through word-of-mouth effects. The expected number of influenced
users I(x) and the cost of the products C(x) for a vector xof seeds are monotone
k-submodular functions [13, 14]. We want to select a vector x= (X1, . . . , Xk)
that contains ksubsets of seeds, so that each subset is given a different product
to advertise and xmaximizes the ratio I(x)
C(x), or equivalently minimizes C(x)
I(x).
3. Coupled feature selection: Consider a set of features Vwhich leads to an accu-
rate pattern classifier. We want to predict kvariables Z1,. . .,Zkusing features
Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions 3
Y1,. . .,Y|V|. Due to communication constraints [16], each feature can be used
to predict only one Zi. When Y1,...Y|V|are pairwise independent given Z, the
monotone k-submodular function F(x) = H(X1, . . . , Xk)Pi[k]PjXiH(Yj|Zi),
where His the entropy function and Xiis a group of features, captures the joint
quality of feature groups [16], and the monotone k-submodular function C(x)
captures their cost. The problem is to select xthat minimizes the ratio C(x)
F(x).
The RS-1 minimization problem has been studied in [1, 15, 17] and its ap-
plications include maximizing the F-measure in information retrieval, as well as
maximizing normalized cuts and ratio cuts [1]. However, the algorithms in these
works cannot be directly applied to our problem.
Contributions. Our contributions can be summarized as follows:
1. We define RS-kminimization problem, a new optimization problem that seeks
to minimize the ratio of two monotone k-submodular functions and finds appli-
cations in influence maximization, sensor placement, and feature selection.
2. We introduce three greedy based approximation algorithms for the problem: k-
GreedRatio,k-StochasticGreedRatio, and Sandwich Approximation Ra-
tio (SAR). The first two algorithms have an approximation ratio that depends
on the curvature of fand the size of the optimal solution. These algorithms
generalize the result of [15] to k-submodular functions and improve the result
of [1] for the RS-1 minimization problem. k-Stochastic-GreedRatio differs
from k-GreedRatio in that it is more efficient, as it uses sampling, and in that
it achieves the guarantee of k-GreedRatio with a probability of at least 1 δ,
for any δ > 0. SAR has an approximation ratio that depends on the values of f,
and it is based on the sandwich approximation strategy [10] for non-submodular
maximization, which we extend to a ratio of k-submodular functions.
3. We experimentally demonstrate the effectiveness and efficiency of our algo-
rithms on the sensor selection problem and the influence maximization problem
outlined above, by comparing them to three baselines. The solutions of our al-
gorithms have several times higher quality compared to those of the baselines.
2 Related Work
The concept of k-submodular function was introduced in [6], and the problem
of maximizing a k-submodular function has been considered in [6, 7, 14, 18]. For
example, [7] studied unconstrained k-submodular maximization and proposed a
k
2k1-approximation algorithm for monotone functions and a 1
2-approximation
algorithm for nonmonotone functions. The work of [14] studied constrained k-
submodular maximization, with an upper bound on the solution size, and pro-
posed k-Greedy-T S , a 1
2-approximation algorithm for monotone functions, and
a randomized version of it. These algorithms cannot deal with our problem.
When k= 1, the RS-kminimization problem coincides with the submod-
ular ratio minimization problem studied in [1, 15, 17]. The work of [1] initi-
ated the study of the latter problem and proposed GreedRatio, an 1
1eκf1-
approximation algorithm, where kfis the curvature of a submodular function f
4 Hau Chan et al.
[9]. The work of [15] provided an improved approximation of |X|
1+(|X|−1)(1ˆκf(X))
for GreedRatio, where |X|is the size of an optimal solution Xand ˆκfis
an alternative curvature notion of f[9]. The work of [17] considered the sub-
modular ratio minimization problem where both of the functions may not be
submodular and showed that GreedRatio provides approximation guarantees
that depend on the submodularity ratio [5] and the curvatures of the functions.
These previous results do not apply to RS-kminimization with k > 1.
3 Preliminaries
We may use the word dimension when referring to a particular subset of x
(k+ 1)V. We use 0= (X1={}, . . . , Xk={}) to denote the vector of kempty
subsets and x= (Xi,xi) to highlight the set Xiin dimension iand xithe
subsets of other dimensions except i.
We define the marginal gain of a k-submodular function fwhen adding
an element to a dimension i[k] to be u,if(x) = f(X1, . . . , Xi1, Xi
{u}, Xi+1, . . . , Xk)f(X1, . . . , Xk), where x(k+ 1)Vand u6∈ Sl[k]Xl.
Thus, we can also define k-submodular functions as those that are monotone
and have a diminishing returns property in each dimension [18].
Definition 1 (k-submodular function). A function f: (k+ 1)VRis k-
submodular if and only if: (a) u,if(x)u,if(y), for all x,y(k+ 1)Vwith
xy,u /∈ ∪`[k]Yi, and i[k], and (b) u,i f(x) + u,jf(x)0, for any
x(k+ 1)V,u /∈ ∪`[k]Xi, and i, j [k]with i6=j.
Part (a) of Definition 1 is known as the diminishing returns property and
part (b) as pairwise monotonicity. We define a k-modular function as follows.
Definition 2 (k-modular function). A function fis k-modular if and only
if u,if(x) = u,i f(y), for all x,y(k+ 1)Vwith xy,u /∈ ∪`[k]Yi, and
i[k].
Without loss of generality, we assume that the functions fand gin RS-k
minimization are normalized such that g(0) = f(0) = 0. Moreover, we assume
that, for each uV, there are dimensions i, i0[k] such that f({u}i,0i)>
0 and g({u}i0,0i0)>0. Otherwise, we can remove each uVsuch that
g({u}i,0i) = 0 for all i[k] from the candidate solution of the problem, as
adding it to any dimension will not decrease the value of the ratio. Also, if there
is uVwith f({u}i,0i) = 0 for all i[k], we could add uto the final solution
as it will not increase the ratio. If there is more than one such element, we need
to determine which dimensions to add the elements into, so that gis maximized,
which boils down to solving a k-submodular function maximization problem.
Applying an existing α-approximation algorithm [6, 7, 14, 18] to that problem
would yield an additional approximation multiplicative factor of αto the ratio
of the final solution of our problem. Finally, we assume that the values of fand
gare given by value oracles.
Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions 5
4 The k-GreedRatio Algorithm
We present our first algorithm for the RS-kminimization problem, called k-
GreedRatio. The algorithm iteratively adds an element that achieves the best
marginal gain to the functions of the ratio f/g and terminates by returning
the subsets (created from each iteration) that have the smallest ratio. We show
that k-GreedRatio has a bounded approximation ratio that depends on the
curvature of the function f. We also show that the algorithm yields an optimal
solution when fand gare k-modular.
Algorithm: k-GreedRatio
Input: V,f: (k+ 1)VR0,g: (k+ 1)VR0
Output: Solution x(k+ 1)V
1j0; xj0;RV;S← {}
2while R6={} do
3(u, i)arg min
uR,i[k]
u,if(xj)
u,ig(xj)
4xj+1 (X1,...,Xi∪ {u},...Xk)
5R← {uR|u /Xi,i[k], and i[k] : u,ig(xj+1 )>0}
6SS∪ {xj+1}
7jj+ 1
8xarg min
xjS
f(xj)
g(xj)
9return x
4.1 The Ratio of k-Submodular Functions
Following [4,9], we define the curvature of a k-submodular function of dimension
i[k] for any x(k+ 1)Vas κf,i(xi) = 1 minuV\Sj6=iXj
u,if(V\{u},xi)
f({u},xi),
and κf,i (Xi,xi) = 1 minuXi\Sj6=iXj
u,if(Xi\{u},xi)
f({u},xi). We extend a relaxed
version [9] of the above definition as ˆκf,i (Xi,xi)=1PuXiu,if(Xi\{u},xi)
PuXif({u},xi).
Note that, for a given x(k+ 1)V, ˆκf,i (Xi,xi)κf,i(Xi,xi)κf,i (xi)
when fis monotone submodular in each dimension [9].
Let ˆκmax
f,i (Xi) = max(Xi,xi)(k+1)Vˆκf ,i(Xi,¯
xi). The following lemma (whose
proof easily follows from Lemma 3.1 of [9]) provides an upper bound on the sum
of the individual elements of a given set of elements for a dimension.
Lemma 1. Given any non-negative and monotone k-submodular function f,
x(k+1)Vand i[k],PuXif({u}i,xi)|Xi|
1+(|Xi|−1)(1ˆκmax
f,i (Xi)) f(Xi,xi).
Notice that the inequality in Lemma 1 depends only on Xi. Thus, it holds for
any x(k+ 1)Vas long as Xi=Xi. We now begin to prove the approximation
guarantee of k-GreedRatio.
Let x= (X
1, ..., X
k)arg min06=x(k+1)V
f(x)
g(x)be an optimal solution of
the RS-kminimization problem. Let S(x) = {x(k+ 1)V| |Xi|= [|X
i|>
6 Hau Chan et al.
0] i[k]}be the subsets of (k+ 1)Vin which each dimension contains at most
one element (with respect to x) where [·] is an indicator function. Given S(x),
we let x0= (X0
1, ..., X0
k)arg min06=xS(x)
f(x)
g(x). We first compare x0with x
using a proof idea from [15].
Theorem 1. Given two non-negative and monotone k-submodular functions f
and g, we have f(x0)
g(x0)αf(x)
g(x), where α=Qi[k]s.t.|X
i|>0
|X
i|
1+(|X
i|−1)(1ˆκmax
f,i (X
i)) .
Proof. We have that
g(x) = g(X
1, .., X
k)X
uiX
i
g({ui}i,x
i)X
u1X
1,...,ukX
k
g({u1}1, ..., {uk}k)
X
u1X
1,...,ukX
k
f({u1}1, ..., {uk}k)g(x0)
f(x0)
Y
i[k]s.t.|X
i|>0
|X
i|
1+(|X
i| − 1)(1 ˆκmax
f,i (X
i))
g(x0)
f(x0)f(x),
where the first inequality is from applying Definition 1(a) to dimension i, the sec-
ond inequality is from applying Definition 1(a) to other dimensions successively,
the third inequality is from noting that f(x0)
g(x0)f(x)
g(x)g(x)f(x)g(x0)
f(x0)
for any xS(x) and summing up the corresponding terms, and the fourth
inequality is from applying Lemma 1 repeatedly from i= 1 to k.ut
Notice that αin Theorem 1 could be hard to compute3. However, the bound
is tight. For instance, if fis k-modular, then x0yields an optimal solution.
Theorem 1 shows that we can approximate the optimal solution, using the
optimal solution x0where each dimension contains at most one element. How-
ever, computing any x0cannot be done efficiently for large k. Our next the-
orem shows that k-GreedRatio solution can be used to approximate any
x0, which, in turn can be used to to approximate any x. To begin, we let
xarg minxVmini[k]
f({x}i,0i)
g({x}i,0i)and let ibe the corresponding dimension.
Theorem 2. Given two non-negative and monotone k-submodular functions f
and g, we have f({x}i,0i)
g({x}i,0i)kf(x00)
g(x00), for any x00 S={x(k+ 1)V| |Xi| ≤
1i[k]}.
Proof. We have that
g(x00) = g({x00
1}, ..., {x00
k})g({x00
1},01) + ... +g({x00
k},0k)
[f({x00
1},01) + ... +f({x00
k},0k)] g({x}i,0i)
f({x}i,0i)kf (x00)g({x}i,0i)
f({x}i,0i),
3It is possible to obtain a computable bound by redefining the curvature related
parameters with respect to xi=0i. The proof in Theorem 1 follows similarly up
until the third inequality. However, the achieved approximation essentially depends
on the product of the sizes of the sets (without all kbut one denominator term).
Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions 7
where the first inequality is from applying the definition of k-submodularity ac-
cording to Inequality 1 in Section 1 repeatedly, the second inequality is from not-
ing that f({x}i,0i)
g({x}i,0i)f({u}j,0j)
g({u}j,0j)g({u}j,0j)f({u}j,0j)g({x}i,0i)
f({x}i,0i)
for any uVand j[k] and summing up the corresponding terms, and the
third inequality is due to monotonicity. ut
Combining Theorems 1 and 2, we have the following result.
Theorem 3. Given two non-negative and monotone k-submodular functions f
and g,k-GreedRatio finds a solution that is at most times of the optimal
solution, where α=Qi[k]s.t.|X
i|>0
|X
i|
1+(|X
i|−1)(1ˆκmax
f,i (X
i)) , in O(|V|2k)time,
assuming it is given (value) oracle access to fand g.
Proof. Let xbe the output of k-GreedRatio. We have that f(x)
g(x)f({x}i,0i)
g({x}i,0i)
kf(x0)
g(x0) f(x)
g(x)where the first inequality holds because xiis the first element
selected by the algorithm, the second inequality is due to Theorem 2 (which
holds for any x00 S), and the third inequality is due to Theorem 1 and α=
Qi[k]s.t.|X
i|>0
|X
i|
1+(|X
i|−1)(1ˆκmax
f,i (X
i)) .k-GreedRatio needs O(|V|2k) time, as
step 3 needs O(|V|k) time and the loop in step 2 is executed O(|V|) times. ut
Our result generalizes the result of [15] to k-submodular functions and im-
proves the result of [1] for the RS-1 minimization problem.
4.2 The Ratio of k-Modular Functions
Theorem 4. Given two non-negative and monotone k-modular functions fand
g,k-GreedRatio finds an optimal solution xarg minx0(k+1)V
f(x0)
g(x0). There
is an O(|V|k+|V|log |V|)-time implementation of k-GreedRatio, assuming it
is given (value) oracle access to fand g.
Proof. The proof follows a similar argument to [1]. From the definition of k-
modular function, fand gsatisfy u,if(x) = f({u}i,0i) and u,ig(x) =
g({u}i,0i) for all x(k+ 1)V,u /∈ ∪`[k]X`, and i[k].
As a result, we can provide an (efficient) alternative implementation of k-
GreedRatio by computing Q(u) = mini[k]
f({u}i,0i)
g({u}i,0i)for each uVand
sorting u0sin increasing order of Q(u) (breaking ties arbitrarily).
Without loss of generality, let Q(u1)Q(u2)... Q(un) be such an ordering
and let i1,i2, .., inbe the corresponding dimensions so that f({u1}i1,0i1)
g({u1}i1,0i1)
. . . f({un}in,0in)
g({un}in,0in). It is not hard to see that k-GreedRatio picks the first i
elements according to the ordering (each in the iiteration).
Let xarg minx(k+1)V
f(x)
g(x)and r=f(x)
g(x). There must be some ujV
such that Q(uj)τ, otherwise rcannot be obtained from the elements of x.
Consider the set 06=xτ= (X1, ..., Xk) where Xl={ujV|Q(uj)
τand ij=l}for each l[k]. First note that xτis among the solutions {xi}n
i=1
8 Hau Chan et al.
obtained by k-GreedRatio. Second, we have that τf(xτ)
g(xτ)τ(i.e., each
of the ratio is bounded by τ). Thus, f(xτ)
g(xτ)=τ.
The above implementation takes O(|V|k) and O(|V|log |V|) time to compute
ratios for each element/dimension and to sort the |V|elements, respectively. ut
5k-StochasticGreedRatio
We introduce a more efficient randomized version of k-GreedRatio that uses
a smaller number of function evaluations at each iteration of the algorithm. The
algorithm is linear in |V|and it uses sampling in a similar way as the algorithm
of [12] for submodular maximization. That is, it selects elements to add into x
based on a sufficiently large random sample of Vinstead of V.
Algorithm: k-StochasticGreedRatio
Input: V,f: (k+ 1)VR0,g: (k+ 1)VR0,δ > 0
Output: Solution x(k+ 1)V
1j0; xj0;RV;S← {}
2while R6={} do
3Qa random subset of size min{llog |V|
δm,|V|}
uniformly sampled with replacement from V\S
4(u, i)arg min
uQ,i[k]
u,if(xj)
u,ig(xj)
5the next steps are the same as steps 4 to 9 of k-GreedRatio
Theorem 5. With probability at least 1δ,k-StochasticGreedRatio out-
puts a solution that is: (a) at most times of the optimal solution when fand
gare non-negative and monotone k-submodular, or (b) optimal when fand g
are non-negative and monotone k-modular where αis the ratio in Theorem 3.
Proof. (a) Let Q1be Qof the first iteration and consider the first element
selected by k-GreedRatio. If |Q1|=|V|, then the probability that the first el-
ement is not contained in Q1is 0. Otherwise, this probability is 11
|V||Q1|
elog |V|
δ=δ
|V|.We have that with probability at least 1δ k-StochasticGreed-
Ratio selects the first element. The claims follows from this and Theorem 3.
(b) It follows from the fact that k-StochasticGreedRatio selects the first
element with probability at least 1 δand Theorem 4. ut
Lemma 2. The time complexity of k-StochasticGreedRatio is O(k|V|log |V|
δ)
for δ|V|
e|V|and O(k|V|2)otherwise, where eis the base of the natural logarithm.
Proof. Step 4 needs O(kmin{dlog |V|
δe,|V|}) = O(kmin{log |V|
δ,|V|}) time and
it is executed O(|V|) times, once per iteration of the loop in step 2. If the
sample size min{dlog |V|
δe,|V|} =dlog |V|
δe, or equivalently if δ|V|
e|V|, then the
algorithm takes O(k|V|log |V|
δ) time. Otherwise, it takes O(k|V|2) time. ut
Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions 9
6 Sandwich Approximation Ratio (SAR)
We present SAR, a greedy based algorithm that provides an approximation
guarantee based on the value of f, by extending the idea of [10] from non-
submodular function maximization to RS-kminimization problems. SAR uses
an upper bound k-submodular function and a lower bound k-submodular func-
tion of the ratio function f/g, applies k-Greedy-TS [14] with size constraint
|V|using each bound function as well as f/g, and returns the solution that
maximizes the ratio among the solutions constructed by k-Greedy-TS4.
SAR uses the functions g(x)
2cand g(x)
c0, where 2cc= maxx(k+1)Vf(x)
and c0= minxVmini[k]f({x}i,0i). It is easy to see that these functions
bound h(x) = g(x)
f(x)from below and above, respectively. While ccannot be
computed exactly, we have 2cc, where cis the solution of the k-Greedy-TS
1
2-approximation algorithm for maximizing a monotone k-submodular function
[14], when applied with function fand solution size threshold |V|.
Algorithm: Sandwich Approximation Ratio (SAR)
Input: V,f: (k+ 1)VR0,g: (k+ 1)VR0,c,c0
Output: Solution xSAR (k+ 1)V
1(x1
`,...,x|V|
`)k-Greedy-TS with g/2cand threshold |V|
2(x1
h,...,x|V|
h)k-Greedy-TS with h=g/f and threshold |V|
3(x1
u,...,x|V|
u)k-Greedy-TS with g/c0and threshold |V|
4return xSAR arg maxx∈{x1
`,...,x|V|
`,x1
h,...,x|V|
h,x1
u,...,x|V|
u}
g(x)
f(x)
Algorithm: k-Greedy-TS
Input: f: (k+ 1)VR0, solution size threshold B
Output: Vector of solutions x1
f,...,xB
f5
1x0
f0
2for j= 1 to Bdo
3(u, i)arg maxuV\R,i[k]u,if(xf)
4xj
fxj1
f
5xj
f(u)i
6RR∪ {u}
7return (x1
f,...,xB
f)
To provide SAR’s approximation guarantee, we define `(x) = g(x)
2c,h(x) =
g(x)
f(x), and u(x) = g(x)
c0for all x(k+ 1)V. Let s(x) = Pi[k]|Xi|be the size of
any x(k+1)V. Let x
harg maxx(k+1)V
g(x)
f(x)be the optimal solution and s=
s(x
h) be the size of the optimal solution. We let xj
`arg maxx(k+1)V:s(x)=j`(x),
xj
harg maxx(k+1)V:s(x)=j
g(x)
f(x), and xj
uarg maxx(k+1)V:s(x)=ju(x).
4SAR can be easily modified to use other algorithms for monotone k-submodular
maximization instead of k-Greedy-TS, such as the algorithm of [7].
5We modify k-Greedy-T S to return every partial solution xj
f, instead of only xB
f.
10 Hau Chan et al.
Theorem 6. Given two non-negative and monotone k-submodular functions f
and g, SAR finds a solution at most 2.max( c0
f(xs
u),f(xs
h)
2c)times the optimal
solution in O(|V|2k)time, assuming it is given (value) oracle access to fand g.
Proof. We first show that, for each j[|V|], maxxj∈{xj
`,xj
h,xj
u}
g(xj)
f(xj)from SAR
approximates g(xjh)
f(xjh). Since s[|V|] and xSAR arg maxxj∈{xj
`,xj
h,xj
u}j[|V|]
g(xj)
f(xj),
our claimed approximation follows immediately. For a fixed j[|V|], we have
h(xj
u) = h(xj
u)
u(xj
u)u(xj
u)h(xj
u)
u(xj
u)
1
2u(xj
u)h(xj
u)
u(xj
u)
1
2u(xj
h)h(xj
u)
u(xj
u)
1
2h(xj
h),
where the first inequality is due to the use of k-Greedy-TS in [14] for a fixed
size j, the second inequality follows from the definition of xj
u, and the third
inequality is from the fact that uupper-bounds h.
We also have h(xj
`)`(xj
`)1
2`(xj
`)1
2`(xj
h)`(xj
h)
h(xj
h)
1
2h(xj
h), where the
first inequality is due to the use of k-Greedy-TS in [14] for a fixed size j, the
second inequality follows from the definition of xj
`, and the third inequality is
from the fact that `lower-bounds h.
From combining h(xj
u)h(xj
u)
u(xj
u)
1
2h(xj
h) and h(xj
`)`(xj
h)
h(xj
h)
1
2h(xj
h), we have
max
xj∈{xj
`,xj
h,xj
u}
h(xj)max h(xj
u)
u(xj
u),`(xj
h)
h(xj
h)!1
2h(xj
h).
From the above for each jand step 4 of SAR, we have
h(xSAR)max
j[|V|](max h(xj
u)
u(xj
u),`(xj
h)
h(xj
h)!1
2h(xj
h))h(xs
u)
u(xs
u),`(xs
h)
h(xs
h)1
2h(xs
h).
It follows that: f(xSAR )
g(xSAR)2max c0
f(xs
u),f(xs
h)
2carg minx(k+1)V
f(x)
g(x). The time
complexity of SAR follows from executing k-Greedy-T S three times. ut
7 Experimental Results
We experimentally evaluate the effectiveness and efficiency of our algorithms for
cost-effective variants [13] of two problems on two publicly available datasets;
a sensor placement problem where sensors have kdifferent types [14], and an
influence maximization problem under the k-topic independent cascade model
[14]. We compare our algorithms to three baseline algorithms, alike those in [14],
as explained below. We implemented all algorithms in C++ and executed them
on an Intel Xeon @ 2.60GHz with 128GB RAM. Our source code and the datasets
we used are available at: https://bitbucket.org/grigorios_loukides/ksub.
Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions 11
7.1 Sensor placement
Entropy and cost functions. We first define the entropy function for the
problem, following [14]. Let the set of random variables ={X1, . . . , Xn}and
H(S) = Psdom SP r[s]·log P r[s] be the entropy of a subset S ⊆ and
dom Sis the domain of S. The conditional entropy of after having observed
Sis H(| S) = H()H(S). Thus, an Swith large entropy H(S) has
small uncertainty and is preferred. In our sensor placement problem, we want
to select locations at which we will install sensors of ktypes, one sensor per
selected location. Let ={Xu
i}i[k],uVbe the set of random variables for each
sensor type i[k] and each location uV. Each Xu
iis the random variable
representing the observation collected from a sensor of type ithat is installed at
location u. Thus, Xi={Xu
i} ⊆ is the set representing the observations for
all locations at which a sensor of type i[k] is installed. Then, the entropy of
a vector x= (X1, . . . , Xk)(k+ 1)Vis given by H(x) = H(i[k]Xi).
Let cibe the cost of installing any sensor of type i[k]. We selected each
ciuniformly at random from [1,10], unless stated otherwise, and computed the
cost of a vector x= (X1, . . . , Xk)(k+ 1)Vusing the cost function C(x) =
Pi[k]ci· |Xi|β, where β(0,1] is a user-specified parameter, similarly to [8].
This function models that the cost of installing an extra sensor of any type i
diminishes when more sensors of that type have been installed (i.e., when |Xi|
is larger) and that the total cost of installing sensors is the sum of the costs of
installing all sensors of each type. The function |Xi|β, for β(0,1], is monotone
submodular, as a composition of a monotone concave function and a monotone
modular function [3]. Thus, C(x) is monotone k-submodular, as a composition of
a monotone concave and a monotone k-modular function (the proof easily follows
from Theorem 5.4 in [3]). The RS-kminimization problem is to minimize C(x)
H(x).
We solve the equivalent problem of maximizing H(x)
C(x).
Setup. We evaluate our algorithms on the Intel Lab dataset (http://db.csail.
mit.edu/labdata/labdata.html) which is preprocessed as in [14]. The dataset
is a log of approximately 2.3 million values that are collected from 54 sensors
installed in 54 locations in the Intel Berkeley research lab. There are three types
of sensors. Sensors of type 1, 2, and 3 collect temperature, humidity, and light
values, respectively. Our k-submodular functions take as argument a vector: (1)
x= (X1) of sensors of type 1, when k= 1; (2) x= (X1, X2) of sensors of type
1 and of type 2, when k= 2, or (3) x= (X1, X2, X3) of sensors of type 1 and of
type 2 and of type 3, when k= 3.
We compared our algorithms to two baselines: 1. Single(i), which allocates
only sensors of type ito locations, and 2. Random, which allocates sensors of
any type randomly to locations. The baselines are similar to those in [14]; the
only difference is that Single(i) is based on H
C. That is, Single(i) adds into
the dimension iof vector xthe location that incurs the maximum gain with
respect to H
C. We tested all different i’s and report results for Single(1), which
performed the best. Following [14], we used the lazy evaluation technique [11]
in k-GreedRatio and SAR, for efficiency. For these algorithms, we maintain
12 Hau Chan et al.
an upper bound on the gain of adding each uin Xi, for i[k], with respect to
H
Cand apply the technique directly. For k-StochasticGreedRatio, we used
δ= 101(unless stated otherwise) and report the average over 10 runs.
(a) (b) (c)
(d) (e) (f)
Fig. 1: (a) Entropy to cost ratio H
Cfor varying k[1,3] and β= 0.1. (b) Entropy
Hfor varying k[1,3] and β= 0.1. (c) Cost Cfor varying k[1,3] and β= 0.1.
(d) Entropy to cost ratio H
Cfor varying k[1,3] and β= 0.9. (b) Entropy H
for varying k[1,3] and β= 0.9. (c) Cost Cfor varying k[1,3] and β= 0.9.
Fig. 2: Entropy to cost ratio for varying
β[0.05,0.9] and k= 2.
Results. Fig. 1 shows that our algo-
rithms outperform the baselines with
respect to entropy to cost ratio. In
these experiments, we used c1=c2=
c3= 1. Specifically, our algorithms
outperform the best baseline, Sin-
gle(1), by 5.2, 5.1, and 3.6 times on
average over the results of Fig. 1a and
1d. Our algorithms perform best when
Cis close to being k-modular (i.e., in
Figs. 1d, 1e and 1f where β= 0.9).
In this case, they outperform Single(1) by at least 6.2 times. The cost func-
tion C(x) increases as βincreases and thus it affects the entropy to cost ratio
H(x)/C(x) more substantially when β= 0.9. Yet, our algorithms again selected
sensors with smaller costs than the baselines (see Fig. 1f), achieving a solution
with much higher ratio (see Fig. 1d). The good performance of k-GreedRatio
and k-StochasticGreedRatio when β= 0.9 is because C(x) is “close” to
k-modular (it is k-modular with β= 1) and these algorithms offer a better
approximation guarantee for a k-modular function (see Theorem 3).
Fig. 2 shows that our algorithms outperform the baselines with respect
to entropy to cost ratio for different values of β[0.05,0.9]. Specifically, k-
GreedRatio,k-StochasticGreedRatio, and SAR outperform the best base-
Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions 13
line, Single(1), by 3.4, 3.1, and 3 times on average, respectively. Also, note that
k-GreedRatio and k-StochasticGreedRatio perform very well for β > 0.5,
as they again were able to select sensors with smaller costs.
Fig. 3a shows the number of function evaluations (H
C,Hand Cfor SAR
and H
Cfor all other algorithms) when kvaries in [1,3]. The number of function
evaluations is a proxy for efficiency and shows the benefit of lazy evaluation [14].
As can be seen, the number of function evaluations is the largest for SAR, since
it evaluates both h=H
Cand g=C, while it is zero for Random, since it
does not evaluate any function to select sensors. Single(1) performs a small
number of evaluations of H
C, since it adds all sensors into a fixed dimension.
k-StochasticGreedRatio performs fewer evaluations than k-GreedRatio
due to the use of sampling. Fig. 3b shows the runtime of all algorithms for
the same experiment as that of Fig. 3a. Observe that all algorithms take more
time as kincreases and that our algorithms are slower than the baselines, since
they perform more function evaluations. However, the runtime of our algorithms
increases sublinearly with k.k-StochasticGreedRatio is the fastest, while
SAR is the slowest among our algorithms.
Figs. 3c and 3d show the impact of parameter δon the entropy to cost ratio
and on runtime of k-StochasticGreedRatio, respectively. As can be seen,
when δincreases, the algorithm finds a slightly worse solution but runs faster.
This is because a smaller δleads to a smaller sample size. In fact, the sample
size was 30% for δ= 105and 10.9% for δ= 0.2.
(a) (b) (c) (d)
Fig. 3: (a) Number of evaluations of H
C,H, or C, for varying k[1,3] and
β= 0.9. (b) Runtime (sec) for varying k[1,3] and β= 0.9. (c) Entropy to
cost ratio H
Cfor varying δ[105,0.2] used in k-StochasticGreedRatio. (d)
Runtime (sec) for varying δ[105,0.2] used in k-StochasticGreedRatio.
7.2 Influence maximization
Influence and cost functions. In the k-topic independent cascade model [14],
kdifferent topics spread through a social network independently. At the ini-
tial time t= 0, there is a vector x= (X1, . . . , Xk) of influenced users called
seeds. Each seed uin Xi,i[k], is influenced about topic iand has a single
chance to influence its out-neighbor v, if vis not already influenced. The node
vis influenced at t= 1 by uon topic iwith probability pi
u,v. When vbecomes
influenced, it stays influenced and has a single chance to influence each of its
out-neighbors that is not already influenced. The process proceeds similarly un-
til no new nodes are influenced. The expected number of influenced users is
I(x) = E[| ∪i[k]Ai(Xi)|], where Ai(Xi) is a random variable representing the
14 Hau Chan et al.
set of users influenced about topic ithrough Xi. The influence function Iis
shown to be k-submodular [14]. The selection of a node uas seed in Xiincurs
a cost C(u, i), which we selected uniformly at random from [2000,20000]. The
cost of xis C(x) = Pu∈∪i[k]XiC(u, i)β
, where β(0,1]. C(x) is mono-
tone k-submodular, as a composition of a monotone concave and a monotone
k-modular function (the proof easily follows from Theorem 5.4 in [3]). The RS-k
minimization problem is to minimize C(x)
I(x). We solve the equivalent problem of
maximizing I(x)
C(x).
(a) (b)
Fig. 4: Spread to cost ratio I
Cfor varying: (a) k[2,6] and β= 0.5, (b) k[2,6]
and β= 0.9
Setup. We evaluate our algorithms on a dataset of a social news website (http:
//www.isi.edu/~lerman/downloads/digg2009.html) following the setup of [14].
The dataset consists of a graph and a log of user votes for stories. Each node
represents a user and each edge (u, v) represents that user ucan watch the ac-
tivity of v. The edge probabilities pi
u,v for each edge (u, v) and topic iwere set
using the method of [2]. We compared our algorithms to three baselines [14]:
1. Single(i); 2. Random, and 3. Degree.Single(i) is similar to that used
in Section 7.1 but it is based on I
C. Following [14], we used the lazy evalua-
tion technique [11] on k-GreedRatio,SAR, and k-StochasticGreedRatio.
For the first two algorithms, we applied the technique similarly to Section 7.1.
For k-StochasticGreedRatio, we maintain an upper bound on the gain of
adding each uinto Xi, for i[1, k], w.r.t. H
Cand select the element in Qwith the
largest gain in each iteration. We tested all i’s in Single(i) and report results for
Single(1) that performed best. Degree sorts all nodes in decreasing order of
out-degree and assigns each of them to a random topic. We simulated the influ-
ence process based on Monte Carlo simulation. For k-StochasticGreedRatio,
we used δ= 101and report the average over 10 runs.
Results. Figs. 4a and 4b show that our algorithms outperform all three base-
lines, by at least 15.3, 3.3, and 1.5 times on average for k-GreedRatio,k-
StochasticGreedRatio, and SAR, respectively. The first two algorithms per-
form best when Cis close to being k-modular (i.e., in Fig. 4b where β= 0.9).
This is because Cis k-modular when β= 1 and these algorithms offer a better
approximation guarantee for a k-modular function (see Theorem 3).
Algorithms for Optimizing the Ratio of Monotone k-Submodular Functions 15
Fig. 5a shows that all algorithms perform similarly for β < 0.5. This is
because in these cases Chas a much smaller value than I. Thus, the benefit
of our algorithms in terms of selecting seeds with small costs is not signifi-
cant. For β0.5, our algorithms substantially outperformed the baselines, with
k-GreedRatio and k-StochasticGreedRatio performing better as βap-
proaches 1 for the reason mentioned above.
(a) (b)
Fig. 5: (a) Spread to cost ratio for varying β[0.05,0.9] and k= 6. (b) Number
of evaluations of I
C,Ior C, for varying k[2,6] and β= 0.7.
Fig. 5b shows the number of evaluations of the functions I
Cand Ifor SAR,
and of I
Cfor all other algorithms, when kvaries in [2,6]. The number of evalu-
ations is the largest for SAR, since SAR applies k-Greedy-TS on both h=I
C
and g=C, and zero for Random and Degree, since these algorithms select
seeds without evaluating I
C.Single(1) performs a small number of function
evaluations of I
C, since it adds all nodes into a fixed dimension (i.e., dimen-
sion 1). k-StochasticGreedRatio performs fewer function evaluations than
k-GreedRatio, because it uses sampling. k-StochasticGreedRatio was also
30% faster than SAR on average but 5 times slower than Single(1).
8 Conclusion
In this paper, we studied RS-kminimization, a new optimization problem that
seeks to minimize the ratio of two monotone k-submodular functions. To deal
with the problem, we developed k-GreedRatio,k-StochasticGreedRatio,
and Sandwich Approximation Ratio (SAR), whose approximation ratios depend
on the curvatures of the k-submodular functions and the values of the functions.
We also demonstrated the effectiveness and efficiency of our algorithms by ap-
plying them to sensor placement and influence maximization problems.
Extensions. One interesting question is to consider the RS-kminimization
problem with size constraints, alike those for k-submodular maximization in [14].
The constrained k-minimization problem seeks to select kdisjoint subsets that
minimize the ratio and either all contain at most a specified number of elements,
or each of them contains at most a specified number of elements. Our algorithms
can be extended to tackle this constrained problem.
Another interesting question is to consider RS-kminimization when fand
gare not exactly k-submodular. A recent work [17] shows that the approxima-
16 Hau Chan et al.
tion ratios of algorithm for RS-1 minimization depend on the curvatures and
submodularity ratios [5] of the functions fand g, when fand gare not sub-
modular. A similar idea can be considered for our algorithms, provided that we
extend the notion of submodularity ratio to k-submodular functions.
Acknowledgments
We would like to thank the authors of [14] for providing the code of the baselines.
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Conference Paper
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