semidefinite bounding procedure for solving

Abstract not found

(ε/log(1/ε)) integrality gap for the Max-Cut SDP based on Euclidean space with the Gaussian probability distribution. This shows that the SDP-rounding algorithm of Charikar-Wirth is essentially best possible. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters

Max-cut problem is one of many NP-hard graph theory problems which attracted many researchers over the years. Though there is almost no hope in finding a polynomialtime algorithm for max-cut problem, various heuristics, or combination of optimization and heuristic methods have been developed

We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem

Let G be an undirected graph for which the standard Max-Cut SDP relaxation achieves at least a c fraction of the total edge weight, 1 2 ≤ c ≤ 1. If the actual optimal cut for G is at most an s fraction of the total edge weight, we say that (c, s) is an SDP gap. We define the SDP gap curve GapSDP

Let G be an undirected graph for which the standard Max-Cut SDP relaxation achieves at least a c fraction of the total edge weight, 1 2 ≤ c ≤ 1. If the actual optimal cut for G is at most an s fraction of the total edge weight, we say that (c, s) is an SDP gap. We define the SDP gap curve GapSDP

In this project, we investigated several approximation algorithms for the Max-Cut problem.

Given an undirected graph with edge weights, the MAX-CUT problem consists in finding a partition of the nodes into two subsets, such that the sum of the weights of the edges having endpoints in different subsets is maximized.

This paper presents a smoothing heuristic for an NP-hard combinatorial problem. Starting with a convex Lagrangian relaxation, a pathfollowing method is applied to obtain good solutions while gradually transforming the relaxed problem into the original problem formulated with an exact penalty function. Starting points are drawn using different sampling techniques that use randomization and eigenvectors. The dual point that defines the convex relaxation is computed via eigenvalue optimization using subgradient techniques. The proposed method turns out to be competitive with the most recent ones. The idea presented here is generic and can be generalized to all box-constrained problems where convex Lagrangian relaxation can be applied. Furthermore, to the best of our knowledge, this is the first time that a Lagrangian heuristic is combined with pathfollowing techniques. Key words. semidefinite programming, quadratic programming, combinatorial optimization, non-convex programming, approximation methods and heuristics, pathfollowing, AMS classifications. 90C22, 90C20, 90C27, 90C26, 90C59 2 1