A note on the maximum clique LP relaxation

We study the well-known linear programming relaxation of the maximum clique problem, whose optimal value is known as the fractional clique number . We show that this relaxation exhibits a large integrality gap: there exist graphs on n vertices for which the fractional clique number exceeds the actual clique number by a factor of Θ( n /log 2 n ).

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Publication Details

Journal
Operations Research Letters
Published
2026-02-13
DOI
https://doi.org/10.1016/j.orl.2026.107425
Primary Topic
Complexity and Algorithms in Graphs
Type
article
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article

A note on the maximum clique LP relaxation

Pavel Zakharov
Operations Research Letters
Complexity and Algorithms in Graphs
article

A note on the maximum clique LP relaxation

Pavel Zakharov
article en

Abstract

We study the well-known linear programming relaxation of the maximum clique problem, whose optimal value is known as the fractional clique number . We show that this relaxation exhibits a large integrality gap: there exist graphs on n vertices for which the fractional clique number exceeds the actual clique number by a factor of Θ( n /log 2 n ).

Operations Research LettersVol. 66
TU Dortmund University (DE)
Openalex Percentile: Top 55%
Complexity and Algorithms in Graphs
DOIOpen Access PDFOpenAlex
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