Exponential-Time Approximation of Hard Problems

TL;DR

This paper explores exponential-time approximation algorithms for NP-hard problems, achieving smaller exponential bases and approximation ratios through novel backtracking and transformation techniques.

Contribution

It introduces new exponential-time approximation schemes that trade off between approximation quality and reduced exponential base in the running time.

Findings

A $(4r-1)$-approximation for Bandwidth in $O^*(2^{n/r})$ time.

Transformations from exact exponential algorithms to faster approximation algorithms.

Extension of exact algorithms' applicability to NP-hard problems through approximation schemes.

Abstract

We study optimization problems that are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. Specifically, we focus on Maximum Independent Set, Vertex Coloring, Set Cover, and Bandwidth. In recent years, many researchers design exact exponential-time algorithms for these and other hard problems. The goal is getting the time complexity still of order $O(c^n)$, but with the constant $c$ as small as possible. In this work we extend this line of research and we investigate whether the constant $c$ can be made even smaller when one allows constant factor approximation. In fact, we describe a kind of approximation schemes -- trade-offs between approximation factor and the time complexity. We study two natural approaches. The first approach consists of designing a backtracking algorithm with a small search tree. We present one result of that kind: a $(4r-1)$-approximation of Bandwidth in time $O^*(2^{n/r})$, for any positive integer $r$. The second approach uses general transformations from exponential-time exact algorithms to approximations that are faster but still exponential-time. For example, we show that for any reduction rate $r$, one can transform any $O^*(c^n)$-time algorithm for Set Cover into a $(1+\ln r)$-approximation algorithm running in time $O^*(c^{n/r})$. We believe that results of that kind extend the applicability of exact algorithms for NP-hard problems.