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 $$d\text {-}\textsc {Hitting Set}{}$$ that runs in time $$n^{{{\,\mathrm{O}\,}}{(d^2 + (d / \epsilon ))}}$$ , uses $${{\,\mathrm{O}\,}}{((d^2 + (d / \epsilon ))\log {n})}$$ bits of space, and achieves an approximation ratio of $${{\,\mathrm{O}\,}}{((d / \epsilon ) n^{\epsilon })}$$ for any positive $$\epsilon \le 1$$ and any $$d \in {\mathbb {N}}$$ . In particular, this yields a factor- $${{\,\mathrm{O}\,}}{(\log {n})}$$ approximation algorithm which runs in time $$n^{{{\,\mathrm{O}\,}}{(\log {n})}}$$ and uses $${{\,\mathrm{O}\,}}{(\log ^2{n})}$$ bits of space (for constant d). As a corollary, we obtain similar bounds for $$\textsc {Vertex Cover}{}$$ and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for $$\textsc {Vertex Cover}{}$$ on graphs with maximum degree $$\varDelta$$ , and an algorithm for computing maximal independent sets, both of which run in time $$n^{{{\,\mathrm{O}\,}}{(\varDelta )}}$$ and use $${{\,\mathrm{O}\,}}{(\varDelta \log {n})}$$ bits of space. For the more general $$d\text {-}\textsc {Hitting Set}{}$$ problem, we devise a factor-d approximation algorithm which runs in time $$n^{{{\,\mathrm{O}\,}}{(d{\delta }^2)}}$$ and uses $${{\,\mathrm{O}\,}}{(d {\delta }^2 \log {n})}$$ bits of space on set families where each element appears in at most $$\delta$$ sets. For $$\textsc {Independent Set}{}$$ restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses $${{\,\mathrm{O}\,}}{(\log {n})}$$ bits of space. We also devise a factor- $${{\,\mathrm{O}\,}}{(d^2)}$$ approximation algorithm for $$\textsc {Dominating Set}{}$$ on d-degenerate graphs which runs in time $$n^{{{\,\mathrm{O}\,}}{(\log {n})}}$$ and uses $${{\,\mathrm{O}\,}}{(\log ^2{n})}$$ bits of space. For d-regular graphs, we show how a known randomized factor- $${{\,\mathrm{O}\,}}{(\log {d})}$$ approximation algorithm can be derandomized to run in time $$n^{{{\,\mathrm{O}\,}}{(1)}}$$ and use $${{\,\mathrm{O}\,}}{(\log n)}$$ bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.
Highlights
Introduction and MotivationThis paper examines the classical approximation problems Vertex Cover, Hitting Set and Dominating Set in the RAM model under additional polylogarithmic space constraints
We develop a factor-O((d/ )n ) approximation algorithm for d–Hitting Set which runs in time nOhd2+(d/ ) and uses O (d2 + (d/ )) log n bits of space (Section 3), where ≤ 1 is an arbitrary positive number and d is a fixed positive integer
We derive a factor-O((1/ )n ) approximation algorithm for Vertex Cover that runs in time nO(1/ )) and uses O((1/ ) log n) bits of space
Summary
This paper examines the classical approximation problems Vertex Cover, Hitting Set and Dominating Set in the RAM model under additional polylogarithmic space constraints. We develop a factor-O((d/ )n ) approximation algorithm for d–Hitting Set which runs in time nOhd2+(d/ ) and uses O (d2 + (d/ )) log n bits of space (Section 3), where ≤ 1 is an arbitrary positive number and d is a fixed positive integer. We give a simple factor-2 approximation algorithm for Vertex Cover on graphs with maximum degree ∆ which runs in time nO(∆) and uses O(∆ log n) bits of space (Section 3.1). √ We give a factor-O( n) approximation algorithm for graphs excluding C4 (a cycle on 4 vertices) as a subgraph, which runs in polynomial time and uses O(log n) bits of space (Section 4.1). We show how a known factor-(2d) approximation algorithm for Independent Set on graphs with average degree d can be implemented to run in polynomial time and use O(log n) bits of space (Section 5)
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