Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard. In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 2-approximation and a randomized-approximation algo-

We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity, and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NP-hard in general. For the well-known subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of all these heuristic is based on constructive proofs that two specific conditions are necessary for the effective capacity inequalities to be facet defining. The proofs show precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 small and medium size problems.

We study the two-level uncapacitated facility location (TUFL) problem. Given two types of facilities, which we call y-facilities and z-facilities, the problem is to decide which facilities of both types to open, and to which pair of y- and z-facilities each client should be assigned, in order to satisfy the demand at maximum profit. We first present two multi-commodity flow formulations of TUFL and investigate the relationship between these formulations and similar formulations of the one-level uncapacitated facility location (UFL) problem. In particular, we show that all nontrivial facets for UFL define facets for the two-level problem, and derive conditions when facets of TUFL are also facets for UFL. For both formulations of TUFL, we introduce new families of facets and valid inequalities and discuss the associated separation problems. We also characterize the extreme points of the LP-relaxation of the first formulation. While the LP-relaxation of a multi-commodity formulation provi...

Submodular-function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximum-entropy sampling, and maximum facility-location problems. Our main result is that for any k ≥ 2 and any ε> 0, there is a natural local-search algorithm which has approximation guarantee of 1/(k + ε) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves a 1/(k + 1)-approximation of Nemhauser, Wolsey and Fisher, obtained more than 30 years ago. Also, our analysis can be applied to the problem of maximizing a linear objective function and even a general non-monotone submodular function subject to k matroid constraints. We show that in these cases the approximation guarantees of our algorithms are 1/(k − 1 + ε) and 1/(k + 1 + 1/k + ε), respectively.

The uncapacitated facility location problem in the following formulation is considered: max S`I Z(S) = X j2J max i2S b ij \Gamma X i2S c i ; where I and J are finite sets, and b ij , c i 0 are rational numbers. Let Z denote the optimal value of the problem and ZR = P j2J min i2I b ij \Gamma P i2I c i . Cornuejols, Fisher and Nemhauser (1977) prove that for the problem with the additional cardinality constraint jSj K, a simple greedy algorithm finds a feasible solution S such that (Z(S)\GammaZ R )=(Z \GammaZ R ) 1\Gammae \Gamma1 0:632. We suggest a polynomial-time approximation algorithm for the unconstrained version of the problem, based on the idea of randomized rounding due to Goemans and Williamson (1994). It is proved that the algorithm delivers a solution S such that (Z(S) \Gamma ZR )=(Z \Gamma ZR ) 2( p 2 \Gamma 1) 0:828. We also show that there exists " ? 0 such that it is NP-hard to find an approximate solution S with (Z(S) \Gamma ZR )=(Z ...

In some industries, a certain part can be needed in a very large number of different configurations. This is the case, e.g., for the electrical wirings in european car factories. Fortunately, a given configuration can be replaced by a more complete, therefore also more expensive, one. The diversity management problem consists in choosing an optimal set of some given number $k$ of configurations that will be produced, any non produced configuration being replaced by the cheapest produced one compatible with it. We model the problem as an integer linear program close to the one commonly used for the $k$-median problem. Our aim is to solve those problems to optimality. The large scale instances we are interested in lead to difficult LP relaxations, which seem to be intractable by the best direct methods currently available. Most of this paper deals with the use of Lagrangean Relaxation to reduce the size of the problem in order to be able subsequently to solve it to optimality via classical integer optimization.

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure. The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with constant capacity K for all plants. These facet inequalities depend on K and thus differ fundamentally from the valid inequalities for the uncapacitated version of the problem. We also introduce a second formulation for a model with indivisible cus-tomer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.

Given a large un-transcribed corpus of speech utterances, we address the problem of how to select a good subset for wordlevel transcription under a given fixed transcription budget. We employ submodular active selection on a Fisher-kernel based graph over un-transcribed utterances. The selection is theoretically guaranteed to be near-optimal. Moreover, our approach is able to bootstrap without requiring any initial transcribed data, whereas traditional approaches rely heavily on the quality of an initial model trained on some labeled data. Our experiments on phone recognition show that our approach outperforms both average-case random selection and uncertainty sampling significantly.

The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part I of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we ...

The cost structures for resource allocation in many network design problems obey economies of scale, meaning that the cost per unit resource becomes cheaper as the amount of resources allocated increases. For instance, if we are purchasing cables to route data in a network, the cost per unit bandwidth reduces as the bandwidth we need to route increases. Another feature of resource allocation is granularity, meaning that the resource can only be purchased in multiples of a certain minimum quantity. Again, in the context of purchasing cables in a network, the minimum capacity cable available might be a T1 line with capacity 1 Mbps. In this