Mildly Exponential Time Approximation Algorithms for Vertex Cover, Uniform Sparsest Cut and Related Problems

TL;DR

This paper develops mildly exponential time approximation algorithms for Vertex Cover and related problems by leveraging the structure of hard instances and the Sum-of-Squares hierarchy, achieving improved trade-offs between running time and approximation quality.

Contribution

It introduces a framework that adapts the Sum-of-Squares hierarchy to produce faster approximation algorithms with better trade-offs for NP-hard problems.

Findings

Vertex Cover approximation improved over previous algorithms.

New algorithms for Sparsest Cut and related problems with better running time.

Trade-off between approximation ratio and exponential running time established.

Abstract

In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the `hard' instances of the Arora-Rao-Vazirani lemma [JACM'09], we show that the Sum-of-Squares hierarchy can be adapted to provide `fast', but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here $n$ denote the number of vertices of the graph and $r$ can be any positive real number greater than 1 (possibly depending on $n$). (i) A $\left(2 - \frac{1}{O(r)}\right)$-approximation algorithm for Vertex Cover that runs in $\exp\left(\frac{n}{2^{r^2}}\right)n^{O(1)}$ time. (ii) An $O(r)$-approximation algorithms for Uniform Sparsest Cut, Balanced Separator, Minimum UnCut and Minimum 2CNF Deletion that runs in $\exp\left(\frac{n}{2^{r^2}}\right)n^{O(1)}$ time. Our algorithm for Vertex Cover improves upon Bansal et al.'s algorithm [arXiv:1708.03515] which achieves $\left(2 - \frac{1}{O(r)}\right)$-approximation in time $\exp\left(\frac{n}{r^r}\right)n^{O(1)}$. For the remaining problems, our algorithms improve upon $O(r)$-approximation $\exp\left(\frac{n}{2^r}\right)n^{O(1)}$-time algorithms that follow from a work of Charikar et al. [SIAM J. Comput.'10].