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Does Subset Sum Admit Short Proofs?

Author Michał Włodarczyk

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Michał Włodarczyk
  • University of Warsaw, Poland

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  • Włodarczyk, Michał: Supported by Polish National Science Centre SONATA-19 grant number 2023/51/D/ST6/00155.

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Michał Włodarczyk. Does Subset Sum Admit Short Proofs?. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 58:1-58:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.58

Abstract

We investigate the question whether Subset Sum can be solved by a polynomial-time algorithm with access to a certificate of length poly(k) where k is the maximal number of bits in an input number. In other words, can it be solved using only few nondeterministic bits?
This question has motivated us to initiate a systematic study of certification complexity of parameterized problems. Apart from Subset Sum, we examine problems related to integer linear programming, scheduling, and group theory. We reveal an equivalence class of problems sharing the same hardness with respect to having a polynomial certificate. These include Subset Sum and Boolean Linear Programming parameterized by the number of constraints. Secondly, we present new techniques for establishing lower bounds in this regime. In particular, we show that Subset Sum in permutation groups is at least as hard for nondeterministic computation as 3Coloring in bounded-pathwidth graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • subset sum
  • nondeterminism
  • fixed-parameter tractability

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