{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-np-hardness-approximation-curve-for","title":"On the NP-Hardness Approximation Curve for Max-2Lin(2)","arxiv_id":"2408.04832","date":"2024-08-09","proceeding":null,"authors":["Björn Martinsson"],"abstract":"In the Max-2Lin(2) problem you are given a system of equations on the form $x_i + x_j \\equiv b \\pmod{2}$, and your objective is to find an assignment that satisfies as many equations as possible. Let $c \\in [0.5, 1]$ 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 $c \\geqslant 0.9232$ then $\\frac{1 - s (c)}{1 - c} > 1.48969$, 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 $(k = 4)$.","url_abs":"https://arxiv.org/abs/2408.04832v2","url_pdf":"https://arxiv.org/pdf/2408.04832v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-np-hardness-approximation-curve-for","repo_url":"https://github.com/bjorn-martinsson/np-hardness-of-max-2lin-2","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}