As one of the challenges posed in their paper “Ten Challenges in Propositional Reasoning and Search,” Selman et al. (1997) mention the development of Integer Programming (IP) models and methods for propositional reasoning. Even though it is straightforward to formulate a satisfiability problem as an integer programming model, their concern is that the basic technique used to solve integer progr...

Many inequalities for Mixed-Integer Linear Programs (MILPs) or pure Integer Linear Programs (ILPs) are derived from the Gomory corner relaxation, where all the nonbinding constraints at an optimal LP vertex are relaxed. Computational results show that the corner relaxation gives a good approximation of the integer hull for problems with general-integer variables, but the approximation is less s...

We consider the Ring Loading Problem with integer demand splitting (RLP). The problem is given with a ring network, in which a required traffic requirement between each selected node pair must be routed on it. Each traffic requirement can be routed in both directions of the ring network while splitting each traffic requirement in two directions only by integer is allowed. The problem is to find...

From the previous lecture, we have proved the following lemma. Lemma 1. If the cost vector c t ∈ [−ρ, ρ] n , and T ≥ 4ρ 2 ln n 2 , then average loss 1 T T t=1 p t c t ≤ 1 T T t=1 c t (i) + , ∀i ∈ [n] In this lecture, we are going to see the application of multi-weight algorithm for solving LPs and SDPs. For illustration purpose, we will be working on an LP example (the LP relaxation for set cov...

We develop faster approximation algorithms for Metric-TSP building on recent, nearly linear time approximation schemes for the LP relaxation [Chekuri and Quanrud, 2017a]. We show that the LP solution can be sparsified via cut-sparsification techniques such as those of Benczur and Karger [2015]. Given a weighted graph G with m edges and n vertices, and ǫ > 0, our randomized algorithm outputs wit...

We consider the problem of MaxMin allocation of indivisible goods. There are m items to be distributed among n players. Each player i has a nonnegative valuation pij for an item j, and the goal is to allocate items to players so as to maximize the minimum total valuation received by each player. There is a large gap in our understanding of this problem. The best known positive result is an Õ( √...

We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. We also show that when the LP relaxation has fractional solution then BP algorithm can be used to solve the LP. These ...

This paper discusses the fixed-hub single allocation problem (FHSAP). In this problem, a network consists of hub nodes and terminal nodes. Hubs are fixed and fully connected; each terminal node is connected to a single hub which routes all its traffic. The goal is to minimize the cost of routing the traffic in the network. In this paper, we propose a linear programming (LP)-based rounding algor...

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we add...

In this paper we introduce by means of examples a new technique for formulating compact (i.e. polynomial-size) LP relaxations in place of exponential-size models requiring separation algorithms. In the same vein as a celebrated theorem by Grotschel, Lovasz and Schrijver, we state the equivalence of compact separation and compact optimization. Among the examples used to illustrate our technique,...