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Optimal Length Cutting Plane Refutations of Integer Programs

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Algorithms and Discrete Applied Mathematics (CALDAM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13947))

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Abstract

In this paper, we discuss the computational complexities of determining optimal length refutations of infeasible integer programs (IPs). We focus on three different types of refutations, namely read-once refutations, tree-like refutations, and dag-like refutations. For each refutation type, we are interested in finding the length of the shortest possible refutation of that type. For our purposes, the length of a refutation is equal to the number of inferences in that refutation. The refutations in this paper are also defined by the types of inferences that can be used to derive new constraints. We are interested in refutations with two inference rules. The first rule corresponds to the summation of two constraints and is called the ADD rule. The second rule is the DIV rule which divides a constraint by a positive integer. For integer programs, we study the complexity of approximating the length of the shortest refutation of each type (read-once, tree-like, and dag-like). In this paper, we show that the problem of finding the shortest read-once refutation is NPO PB-complete. Additionally, we show that the problem of finding the shortest tree-like refutation is NPO-hard for IPs. We also show that the problem of finding the shortest dag-like refutation is NPO-hard for IPs. Finally, we show that the problems of finding the shortest tree-like and dag-like refutations are in FPSPACE.

This research was supported in part by the Air-Force Office of Scientific Research through Grant FA9550-19-1-0177 and in part by the Air-Force Research Laboratory, Rome through Contract FA8750-17-S-7007.

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Authors and Affiliations

  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    K. Subramani & P. Wojciechowski

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  1. K. Subramani
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  2. P. Wojciechowski
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Correspondence to K. Subramani .

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Editors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology Delhi, New Delhi, India

    Amitabha Bagchi

  2. Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar, India

    Rahul Muthu

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Subramani, K., Wojciechowski, P. (2023). Optimal Length Cutting Plane Refutations of Integer Programs. In: Bagchi, A., Muthu, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2023. Lecture Notes in Computer Science, vol 13947. Springer, Cham. https://doi.org/10.1007/978-3-031-25211-2_2

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  • DOI: https://doi.org/10.1007/978-3-031-25211-2_2

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