Polynomial time approximation schemes for dense instances of NP-hard problems

Abstract

We present a unified framework for designing polynomial time approximation schemes (PTASs) for``dense'' instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimum k-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degree 0(n), although our algorithms solve most of these problems so long as the average degree is 0(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs, where the objective function and the constraints are``dense'' polynomials of constant degree. ]

Rearrangement of Noisy Genomes

“…Since our compatibility graph is dense, methods such as those introduced in [1] might speed up the MWC search.…”

“…Empirically, it was demonstrated to work better than competing algorithms on denser graphs. In the case of genome comparison, we would expect G to become dense, in the sense of [1], as the number of chromosomes increases, as explained in Section 1.…”

“…This is submitted to a search for a maximum-weight clique (MWC), using the algorithm in [6], where the weights are just the number of genes in the pre-strip. Though of course this algorithm may not execute in polynomial time, it is relatively efficient for dense graphs [1], such as the one representing pre-strips for multichromosomal genomes: here, a given pre-strip will not usually involve the same two chromosomes as another pre-strip, and hence will necessarily be compatible with it.…”

Sampling Sub-problems of Heterogeneous Max-cut Problems and Approximation Algorithms

“…The Max-Cut problem is known to be NP-hard, both for general graphs and when restricted to dense graphs [3], where a graph on n vertices is dense if it contains (n 2 ) edges. Thus, much effort has gone into designing and analyzing approximation algorithms for the Max-Cut problem.…”

“…Work originating with [3] has focused on designing PTASs for a large class of NP-hard optimization problems, such as the Max-Cut problem, when the problem instances are dense [1,3,8,17,18,21]. [3] and [8], using quite different methods, designed approximation algorithms for Max-Cut (and other problems) that achieve an additive error of n 2 (where > 0, ∈ (1) is an error parameter) in a time poly(n) (and exponential in 1/ ); this implies relative error for dense instances of these problems.…”