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

Author Björn Martinsson

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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Constraint and logic programming
  • Theory of computation → Graph algorithms analysis
Keywords
  • Inapproximability
  • NP-hardness
  • 2Lin(2)
  • Max-Cut
  • Gadget

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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).

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.11

Author Details

Björn Martinsson
  • KTH Royal Institute of Technology, Stockholm, Sweden

Supplementary Materials

References

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