Narrow sieves for parameterized paths and packings

TL;DR

This paper introduces randomized algorithms for solving complex combinatorial problems like k-path, p-packing, and q-dimensional p-matching efficiently, with improved exponential bases and polynomial space usage, building on recent advanced techniques.

Contribution

The paper presents new randomized algorithms with smaller exponential bases for key combinatorial problems, improving efficiency over previous methods.

Findings

Algorithms solve problems with high probability in exponential time based on parameters

Exponential bases are significantly reduced compared to prior work

Method extends recent techniques by Koutis, Williams, and Björklund

Abstract

We present randomized algorithms for some well-studied, hard combinatorial problems: the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space; the constant bases of the exponentials are significantly smaller than in previous works. For example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with $d$ colors in time within a polynomial factor of O(2^{(d-1)n/2}). Our techniques build upon and generalize some recently published ideas by I. Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj\"orklund (STACS 2010, FOCS 2010).