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APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.7

A Logarithmic Approximation of Linearly-Ordered Colourings

Authors: Johan Håstad, Björn Martinsson, Tamio-Vesa Nakajima, and Stanislav Živný

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract

A linearly ordered (LO) k-colouring of a hypergraph assigns to each vertex a colour from the set {0,1,…,k-1} in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO k-colouring of an LO 2-colourable 3-uniform hypergraph for any constant k ≥ 2 [STACS'21] but even the case k = 3 is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with O^*(√n) colours [ICALP'22] and an LO colouring with O^*(n^(1/3)) colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with O^*(n^(1/5)) colours. We present two simple polynomial-time algorithms that find an LO colouring with O(log₂(n)) colours, which is an exponential improvement.

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Johan Håstad, Björn Martinsson, Tamio-Vesa Nakajima, and Stanislav Živný. A Logarithmic Approximation of Linearly-Ordered Colourings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 7:1-7:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hastad_et_al:LIPIcs.APPROX/RANDOM.2024.7,
  author =	{H\r{a}stad, Johan and Martinsson, Bj\"{o}rn and Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav},
  title =	{{A Logarithmic Approximation of Linearly-Ordered Colourings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{7:1--7:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.7},
  URN =		{urn:nbn:de:0030-drops-210006},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.7},
  annote =	{Keywords: Linear ordered colouring, Hypergraph, Approximation, Promise Constraint Satisfaction Problems}
}

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APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.11

On the NP-Hardness Approximation Curve for Max-2Lin(2)

Authors: Björn Martinsson

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract

In the Max-2Lin(2) problem you are given a system of equations on the form x_i + x_j ≡ b mod 2, and your objective is to find an assignment that satisfies as many equations as possible. Let c ∈ [0.5, 1] denote the maximum fraction of satisfiable equations. In this paper we construct a curve s (c) such that it is NP-hard to find a solution satisfying at least a fraction s of equations. This curve either matches or improves all of the previously known inapproximability NP-hardness results for Max-2Lin(2). In particular, we show that if c ⩾ 0.9232 then (1 - s(c))/(1 - c) > 1.48969, which improves the NP-hardness inapproximability constant for the min deletion version of Max-2Lin(2). Our work complements the work of O'Donnell and Wu that studied the same question assuming the Unique Games Conjecture. Similar to earlier inapproximability results for Max-2Lin(2), we use a gadget reduction from the (2^k - 1)-ary Hadamard predicate. Previous works used k ranging from 2 to 4. Our main result is a procedure for taking a gadget for some fixed k, and use it as a building block to construct better and better gadgets as k tends to infinity. Our method can be used to boost the result of both smaller gadgets created by hand (k = 3) or larger gadgets constructed using a computer (k = 4).

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Björn Martinsson. On the NP-Hardness Approximation Curve for Max-2Lin(2). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 11:1-11:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{martinsson:LIPIcs.APPROX/RANDOM.2024.11,
  author =	{Martinsson, Bj\"{o}rn},
  title =	{{On the NP-Hardness Approximation Curve for Max-2Lin(2)}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{11:1--11:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.11},
  URN =		{urn:nbn:de:0030-drops-210049},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.11},
  annote =	{Keywords: Inapproximability, NP-hardness, 2Lin(2), Max-Cut, Gadget}
}

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Documents authored by Martinsson, Björn

A Logarithmic Approximation of Linearly-Ordered Colourings

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On the NP-Hardness Approximation Curve for Max-2Lin(2)

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