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DOI: 10.4230/LIPIcs.ITCS.2023.1

Worst-Case to Expander-Case Reductions

Authors: Amir Abboud and Nathan Wallheimer

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract

In recent years, the expander decomposition method was used to develop many graph algorithms, resulting in major improvements to longstanding complexity barriers. This powerful hammer has led the community to (1) believe that most problems are as easy on worst-case graphs as they are on expanders, and (2) suspect that expander decompositions are the key to breaking the remaining longstanding barriers in fine-grained complexity. We set out to investigate the extent to which these two things are true (and for which problems). Towards this end, we put forth the concept of worst-case to expander-case self-reductions. We design a collection of such reductions for fundamental graph problems, verifying belief (1) for them. The list includes k-Clique, 4-Cycle, Maximum Cardinality Matching, Vertex-Cover, and Minimum Dominating Set. Interestingly, for most (but not all) of these problems the proof is via a simple gadget reduction, not via expander decompositions, showing that this hammer is effectively useless against the problem and contradicting (2).

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Amir Abboud and Nathan Wallheimer. Worst-Case to Expander-Case Reductions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abboud_et_al:LIPIcs.ITCS.2023.1,
  author =	{Abboud, Amir and Wallheimer, Nathan},
  title =	{{Worst-Case to Expander-Case Reductions}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.1},
  URN =		{urn:nbn:de:0030-drops-175044},
  doi =		{10.4230/LIPIcs.ITCS.2023.1},
  annote =	{Keywords: Fine-Grained Complexity, Expander Decomposition, Reductions, Exact and Parameterized Complexity, Expander Graphs, Triangle, Maximum Matching, Clique, 4-Cycle, Vertex Cover, Dominating Set}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2022.27

Improved Compression of the Okamura-Seymour Metric

Authors: Shay Mozes, Nathan Wallheimer, and Oren Weimann

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract

Let G = (V,E) be an undirected unweighted planar graph. Let S = {s_1,…,s_k} be the vertices of some face in G and let T ⊆ V be an arbitrary set of vertices. The Okamura-Seymour metric compression problem asks to compactly encode the S-to-T distances. Consider a vector storing the distances from an arbitrary vertex v to all vertices S = {s_1,…,s_k} in their cyclic order. The pattern of v is obtained by taking the difference between every pair of consecutive values of this vector. In STOC'19, Li and Parter used a VC-dimension argument to show that in planar graphs, the number of distinct patterns, denoted p_#, is only O(k³). This resulted in a simple Õ(min{k⁴+|T|, k⋅|T|}) space compression of the Okamura-Seymour metric. We give an alternative proof of the p_# = O(k³) bound that exploits planarity beyond the VC-dimension argument. Namely, our proof relies on cut-cycle duality, as well as on the fact that distances among vertices of S are bounded by k. Our method implies the following: (1) An Õ(p_#+k+|T|) space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to Õ(min{k³+|T|, k⋅|T|}). (2) An optimal Õ(k+|T|) space compression of the Okamura-Seymour metric, in the case where the vertices of T induce a connected component in G. (3) A tight bound of p_# = Θ(k²) for the family of Halin graphs, whereas the VC-dimension argument is limited to showing p_# = O(k³).

Cite as

Shay Mozes, Nathan Wallheimer, and Oren Weimann. Improved Compression of the Okamura-Seymour Metric. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mozes_et_al:LIPIcs.ISAAC.2022.27,
  author =	{Mozes, Shay and Wallheimer, Nathan and Weimann, Oren},
  title =	{{Improved Compression of the Okamura-Seymour Metric}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.27},
  URN =		{urn:nbn:de:0030-drops-173123},
  doi =		{10.4230/LIPIcs.ISAAC.2022.27},
  annote =	{Keywords: Shortest paths, planar graphs, metric compression, distance oracles}
}

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Documents authored by Wallheimer, Nathan

Worst-Case to Expander-Case Reductions

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Improved Compression of the Okamura-Seymour Metric

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