Supermodular Set Function Research Articles

Supermodularity and valid inequalities for quadratic optimization with indicators

AbstractWe study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtained from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.

A comment on performance guarantees of a greedy algorithm for minimizing a supermodular set function on comatroid

We provide a counterexample to the performance guarantee obtained in the paper “Il’ev, V., Linker, N., 2006. Performance guarantees of a greedy algorithm for minimizing a supermodular set function on comatroid”, which was published in Volume 171 of the European Journal of Operational Research. We point out where this error originates from in the proof of the main theorem.

Graph Matching Via the Lens of Supermodularity

Graph matching, the problem of aligning a pair of graphs so as to minimize their edge disagreements, has received widespread attention owing to its broad spectrum of applications in data science. As the problem is NP–hard in the worst-case, a variety of approximation algorithms have been proposed for obtaining high quality, suboptimal solutions. In this article, we approach the task of designing an efficient polynomial-time approximation algorithm for graph matching from a previously unconsidered perspective. Our key result is that graph matching can be formulated as maximizing a monotone, supermodular set function subject to matroid intersection constraints. We leverage this fact to apply a discrete optimization variant of the minorization-maximization algorithm which exploits supermodularity of the objective function to iteratively construct and maximize a sequence of global lower bounds on the objective. At each step, we solve a maximum weight matching problem in a bipartite graph. Differing from prior approaches, the algorithm exploits the combinatorial structure inherent in the problem to generate a sequence of iterates featuring monotonically non-decreasing objective value while always adhering to the combinatorial matching constraints. Experiments on real-world data demonstrate the empirical effectiveness of the algorithm relative to the prevailing state-of-the-art.

QoS-Driven Efficient Client Association in High-Density Software-Defined WLAN

A software-defined wireless local area network (SDWLAN) has gained significant interest recently from both the academic and industrial communities and initiated a paradigmatic reconsideration on the stereotyped management and control of current WLAN due to its flexibility and programmability. However, with the proliferation of mobile devices, efficient client association with quality of service (QoS) guarantees in the high-density SDWLAN is a very challenging issue. In this paper, we study the client association problem in the SDWLAN with new features, including centralized association, global network state awareness, seamless handoff, and flow-level association. We formulate the client association in a high-density scenario as an optimization problem aiming to minimize the interpacket delay of individual flows, based on an unsaturated and heterogeneous Markovian analytical model. Furthermore, we interpret the optimization problem as an NP-hard supermodular set function minimization problem, which is solved by two low-complexity heuristic methods, namely the greedy algorithm and the bounded local search algorithm. Through simulations, we demonstrate the effectiveness of our proposed client association solution.

Partition Constrained Covering of a Symmetric Crossing Supermodular Function by a Graph

We are given a symmetric crossing supermodular set function $p$ on $V$ and a partition $\mathcal{P}$ of $V$. We solve the problem of finding a graph with vertex set $V$ having edges only between the classes of $\mathcal{P}$ such that for every subset $X$ of $V$ the cut of the graph defined by $X$ contains at least $p(X)$ edges. The objective is to minimize the number of edges of the graph. This problem is a common generalization of the global edge-connectivity augmentation of a graph with partition constraints, which was solved by Bang-Jensen et al. [SIAM J. Discrete Math., 12 (1999), pp. 160--207] and the problem of covering a symmetric crossing supermodular set function solved by Benczur and Frank [Math. Program., 84 (1999), pp. 483--503]. Our problem can be considered as an abstract form of the problem of global edge-connectivity augmentation of a hypergraph with partition constraints, which was earlier solved by the authors [J. Graph Theory, 72 (2013), pp. 291--312].

Approximate solution of the p-median minimization problem

A version of the facility location problem (the well-known p-median minimization problem) and its generalization—the problem of minimizing a supermodular set function—is studied. These problems are NP-hard, and they are approximately solved by a gradient algorithm that is a discrete analog of the steepest descent algorithm. A priori bounds on the worst-case behavior of the gradient algorithm for the problems under consideration are obtained. As a consequence, a bound on the performance guarantee of the gradient algorithm for the p-median minimization problem in terms of the production and transportation cost matrix is obtained.

Hub Location as the Minimization of a Supermodular Set Function

This paper highlights how a general class of hub location problems can be modeled as the minimization of a real-valued supermodular set function. Well-known problems such as uncapacitated hub location, p-hub median, and hub arc location, among others, are shown to be particular cases of this class. Two integer programming formulations are introduced and compared. One uses path-based variables, frequently employed in hub location, whereas the other exploits properties of supermodular functions. In addition, several worst case bounds for a greedy and a local improvement heuristic are obtained for the general class and for some particular cases in which sharper bounds can be devised. Computational experiments are performed to compare both formulations when used with a general purpose solver. Computational results obtained on benchmark instances confirm the superiority of the supermodular formulation.

A note on parity constrained orientations

This note extends a result of Frank, Jordan, and Szigeti [3] on parity constrained orientations with connectivity requirements. Given a hypergraph H, a non-negative intersecting supermodular set function p, and a preferred in-degree parity for every node, a formula is given on the minimum number of nodes with wrong in-degree parity in an orientation of H covering p.

Performance guarantees of a greedy algorithm for minimizing a supermodular set function on comatroid

We consider the problem of minimizing a supermodular set function on comatroid whose special case is the well-known NP-hard minimization p-median problem. The main result of the paper is a tight bound on the performance guarantee of a greedy heuristic for this problem. As a corollary an analog of the Rado–Edmonds theorem for comatroids is obtained.

Covering symmetric supermodular functions by uniform hypergraphs

We consider the problem of finding a uniform hypergraph that satisfies cut demands defined by a symmetric crossing supermodular set function. We give min–max formulas for both the degree specified and the minimum cardinality problem. These results include as a special case a formula on the minimum number of r-hyperedges whose addition to an initial hypergraph will make it k-edge-connected.

An approximation guarantee of the greedy descent algorithm for minimizing a supermodular set function

We consider the problem of minimizing a supermodular set function whose special case is the well-known NP-hard p-median problem. The main result of the paper is a tight bound on the approximation ratio of a greedy heuristic (discrete analog of the steepest descent algorithm) for this problem.

Covering symmetric supermodular functions by graphs

The minimum number of edges of an undirected graph covering a symmetric, supermodular set-function is determined. As a special case, we derive an extension of a theorem of J. Bang-Jensen and B. Jackson on hypergraph connectivity augmentation.

Bounding a Probability Measure Over a Polymatroid with an Application to Transportation Problems

For M = {1, …, m} and a supermodular set function f: 2M → ℛ, define the polymatroid P(M, f) = {x: ∑j∈A xj ≥ f(A), A ⊆ M; x ∈ ℛm}. We develop a bound for the probability P{X ∈ P(M, f)}, where X is a random m-vector. In particular we show that if X, X1, …, Xm, are independent and identically distributed nonnegative random variables with new better than used (NBU) distribution function, then P{X ∈ P(M, f)} ≥ P{X ≥ f(M̂)}, where M̂ = arg max {f(A): A ⊆ M}. We apply this result to transportation problems with random supply and/or demand. Suppose we have a set K = {1, …, k} of k supply nodes with a random supply Si, i ∈ K and a set N = {1, …, n} of n demand nodes with a random demand Dj, j ∈ N. Let β be the probability that the random demand can be met by the random supply. If the demand and supply are mutually independent, D1, …, Dn, are independent, and S1, …, Sk, are independent and have NBU distribution function, then β ≤ ∏j=1n P{Zj ≥ Dj}, where Zj = ∑i∈K Si rij, j∈N and rij takes the value one if node i can supply the demand node j and zero otherwise (i ∈ K and j ∈ N).