Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover (min-VC), a type of integer programming (IP) problem. A lattice-gas model on the Erdös-Rényi random graphs of α-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical mechanical model with a one-parameter family. Statistical mechan...

Recently, Chan et al. (2000) proposed an efficient heuristic for a class of finite time planning horizon single-warehouse multi-retailer system with complex (nonlinear) transportation cost structures. Their heuristic is based on solving a large LP relaxation of the underlying problem. In this paper, we analyze the strength of the proposed LP relaxation and show that it attains the best theoreti...

In this section we consider another method for solving the set cover problem approximately. The method uses randomized rounding of the solution obtained from the linear programming (LP) relaxation of set cover. The basic idea behind this algorithm is first to solve the LP relaxation for the set cover problem. We can think of the solution ~x as a probability to either select a set or not. Note t...

The problem of obtaining the maximum a posteriori estimate of a general discrete random field (i.e. a random field defined using a finite and discrete set of labels) is known to be NP-hard. However, due to its central importance in many applications, several approximate algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on conve...

Although the lift-and-project operators of Lovász and Schrijver have been the subject of intense study, their M(K , K ) operator has received little attention. We consider an application of this operator to the stable set problem. We begin with an initial linear programming (LP) relaxation consisting of clique and non-negativity inequalities, and then apply the operator to obtain a stronger ext...

We study the Maximum Flow Network Interdiction Problem (MFNIP). We present two classes of polynomially separable valid inequalities for Cardinality MFNIP. We also prove the integrality gap of the LP relaxation of Wood’s [19] integer program is not bounded by a constant factor, even when the LP relaxation is strengthened by our valid inequalities. Finally, we provide an approximation-factor-pres...

Markov random fields (MRFs) are a powerful tool for modelling statistical dependencies for a set of random variables using a graphical representation. An important computational problem related to MRFs, called maximum a posteriori (MAP) inference, is finding a joint variable assignment with the maximal probability. It is well known that the two popular optimisation techniques for this task, lin...

Loopy belief propagation has been employed in a wide variety of applications with great empirical success, but it comes with few theoretical guarantees. In this paper we investigate the use of the max-product form of belief propagation for weighted matching problems on general graphs. We show that max-product converges to the correct answer if the linear programming (LP) relaxation of the weigh...

Label assignment problems with large state spaces are important tasks especially in computer vision. Often the pairwise interaction (or smoothness prior) between labels assigned at adjacent nodes (or pixels) can be described as a function of the label difference. Exact inference in such labeling tasks is still difficult, and therefore approximate inference methods based on a linear programming ...

Direct linear programming (LP) solution to binary sub-modular MRF energy has recently been promoted because i) the solution is identical to the solution by graph cuts, ii) LP is naturally parallelizable and iii) it is flexible in incorporation of constraints. Nevertheless, the conventional LP relaxation for MRF incurs a large number of auxiliary variables and constraints, resulting in expensive...