We show that solving linear programming (LP) relaxations of many classical NP-hard combinatorial optimization problems is as hard as solving the general LP problem. Precisely, the general LP can be reduced in linear time to the LP relaxation of each of these problems. This result poses a fundamental limitation for designing efficient algorithms to solve the LP relaxations, because finding such ...

We propose a new LP relaxation for obtaining the MAP assignment of a binary MRF with pairwise potentials. Our relaxation is derived from reducing the MAP assignment problem to an instance of a recently proposed Bipartite Multi-cut problem where the LP relaxation is guaranteed to provide an O(log k) approximation where k is the number of vertices adjacent to non-submodular edges in the MRF. We t...

We investigate the question of tightness of linear programming (LP) relaxation for finding a maximum weight independent set (MWIS) in sparse random weighted graphs. We show that an edge-based LP relaxation is asymptotically tight for Erdos-Renyi graph G(n, c/n) for c ≤ 2e and random regular graph G(n, r) for r ≤ 4 when node weights are i.i.d. with exponential distribution of mean 1. We establis...

We study the multiway cut problem in directed graphs and one of its special cases, the node-weighted multiway cut problem in undirected graphs. In DIRECTED MULTIWAY CUT (DIR-MC) the input is an edge-weighted directed graph G = (V,E) and a set of k terminal nodes {s1, s2, . . . , sk} ⊆ V ; the goal is to find a min-weight subset of edges whose removal ensures that there is no path from si to sj ...

In the k-edge-connected spanning subgraph problem we are given a graph (V,E) and costs for each edge, and want to find a minimum-cost F ⊂ E such that (V, F ) is k-edge-connected. We show there is a constant ǫ > 0 so that for all k > 1, finding a (1 + ǫ)-approximation for k-ECSS is NP-hard, establishing a gap between the unit-cost and general-cost versions. Next, we consider the multi-subgraph c...

1.1 Max-flow min-cut A flow network is a directed graph D = (V,E) with two distinguished vertices s and t called the source and the sink, respectively. Moreover, each arc (u, v) ∈ E has a certain capacity c(u, v) ≥ 0 assigned to it. Let X be a proper non-empty subset of V . Let X̄ := V −X , then the pair (X, X̄) forms a partition of V , called a cut of D. The set of arcs of D going from X to X̄ is...

We present a exible user-driven ILP tool for the optimisation component of the TRACS II driver scheduling system. The system allows the user to select from a number of objective functions and to drive the LP relaxation through one of a range of optimisation processes. As a default we provide a Sherali objective which minimises the number of shifts, and within that yields the least cost. The def...

We study the behavior of heuristics based on the LP relaxation with respect to the underlying constraindness of the problem. Our study focuses on the Latin square (or quasigroup) completion problem [1]) as a prototype for highly combinatorial problems. We find that simple techniques based on the LP relaxation of the problem provide satisfactory guidance for underand over-constrained instances. ...

The need to recover a train driver schedule occurs during major disruptions in the daily railway operations. Using data from the train driver schedule of the Danish passenger railway operator DSB S-tog A/S, a solution method to the Train Driver Recovery Problem (TDRP) is developed. The TDRP is formulated as a set partitioning problem. The LP relaxation of the set partitioning formulation of the...

LP relaxation One way to deal with this is to relax the integrality constraints and allow xe ∈ [0, 1] to get a linear program, which can be solved in polynomial-time. However, this gives rise to fractional matchings. Characteristic vectors of matchings in G can be seen as points in R where m = |E|. The convex hull of all the matchings forms a polytope called the matching polytope M. However, th...