[2211.03311] The Complexity of Recognizing Facets for the Knapsack Polytope
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Mathematics > Optimization and Control

arXiv:2211.03311 (math)
[Submitted on 7 Nov 2022 (v1), last revised 24 Sep 2023 (this version, v2)]

Title:The Complexity of Recognizing Facets for the Knapsack Polytope

Authors:Rui Chen, Haoran Zhu
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Abstract:The complexity class DP is the class of all languages that are the intersection of a language in NP and a language in co-NP, as coined by Papadimitriou and Yannakakis (1982). Hartvigsen and Zemel (1992) conjectured that recognizing a facet for the knapsack polytope is DP-complete. While it has been known that the recognition problems of facets for polytopes associated with other well-known combinatorial optimization problems, e.g., traveling salesman, node/set packing/covering, are DP-complete, this conjecture on recognizing facets for the knapsack polytope remains open. We provide a positive answer to this conjecture. Moreover, despite the DP-hardness of the recognition problem, we give a polynomial time algorithm for deciding if an inequality with a fixed number of distinct positive coefficients defines a facet of a knapsack polytope, generalizing a result of Balas (1975).
Subjects: Optimization and Control (math.OC); Computational Complexity (cs.CC)
Cite as: arXiv:2211.03311 [math.OC]
  (or arXiv:2211.03311v2 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.2211.03311

Submission history

From: Haoran Zhu [view email]
[v1] Mon, 7 Nov 2022 05:26:12 UTC (37 KB)
[v2] Sun, 24 Sep 2023 22:11:16 UTC (27 KB)
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