Approximation in (Poly-) Logarithmic Space

TL;DR

This paper introduces new approximation algorithms for classical graph and set problems that operate efficiently within logarithmic space constraints, achieving near-optimal ratios and running times.

Contribution

It presents novel approximation algorithms for various graph and set problems that are optimized for space efficiency, including new bounds for d-Hitting Set, Vertex Cover, and other problems.

Findings

Approximation algorithms with O(log n) space for Vertex Cover and related problems.

Factor-d approximation for d-Hitting Set with space O(d δ^2 log n).

Derandomized algorithms for d-regular graphs with O(log n) space.

Abstract

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for d-Hitting Set that runs in time n^{O(d^2 + d/\epsilon})}, uses O((d^2 + d/\epsilon) log n) bits of space, and achieves an approximation ratio of O((d/{\epsilon}) n^{\epsilon}) for any positive \epsilon \leq 1 and any natural number d. In particular, this yields a factor-O(log n) approximation algorithm which runs in time n^{O(log n)} and uses O(log^2 n) bits of space (for constant d). As a corollary, we obtain similar bounds for Vertex Cover and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for Vertex Cover on graphs with maximum degree \Delta, and an algorithm for computing maximal independent sets which both run in time n^{O(\Delta)} and use O(\Delta log n) bits of space. For the more general d-Hitting Set problem, we devise a factor-d approximation algorithm which runs in time n^{O(d {\delta}^2)} and uses O(d {\delta}^2 log n) bits of space on set families where each element appears in at most \delta sets. For Independent Set restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses O(log n) bits of space. We also devise a factor-O(d^2) approximation algorithm for Dominating Set on d-degenerate graphs which runs in time n^{O(log n)} and uses O(log^2 n) bits of space. For d-regular graphs, we show how a known randomized factor-O(log d) approximation algorithm can be derandomized to run in time n^{O(1)} and use O(log n) bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.