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Approximation in (Poly-) Logarithmic Space

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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.

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  1. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Arindam Biswas, Venkatesh Raman & Saket Saurabh

  2. University of Bergen, Bergen, Norway

    Saket Saurabh

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  1. Arindam Biswas
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Biswas, A., Raman, V. & Saurabh, S. Approximation in (Poly-) Logarithmic Space. Algorithmica 83, 2303–2331 (2021). https://doi.org/10.1007/s00453-021-00826-7

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