Approximation in (Poly-) Logarithmic 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 $$d\text {-}\textsc {Hitting Set}{}$$ 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.