On the Hardness of Compressing Weights

Bart M.P. Jansen, Shivesh K. Roy, Micha Włodarczyk

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices n, is significantly harder than the unweighted problem by presenting an O(n3ϵ) lower bound on the size of the kernel, under the assumption that NP ⊆ coNP/poly. This lower bound is essentially tight: we show that we can reduce the problem to the case with weights bounded by 2O(n), which yields a randomized kernel of O(n3) bits. We generalize these results to the weighted d-Uniform Hyperclique problem, Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems (CSPs). We also study weighted minimization problems and show that weight compression is easier when we only want to preserve the collection of optimal solutions. Namely, we show that for node-weighted Vertex Cover on bipartite graphs it is possible to maintain the set of optimal solutions using integer weights from the range [1, n], but if we want to maintain the ordering of the weights of all inclusion-minimal solutions, then weights as large as 2Ω(n) are necessary.

Original languageEnglish
Title of host publication46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021
EditorsFilippo Bonchi, Simon J. Puglisi
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772013
DOIs
StatePublished - 1 Aug 2021
Externally publishedYes
Event46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021 - Tallinn, Estonia
Duration: 23 Aug 202127 Aug 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume202
ISSN (Print)1868-8969

Conference

Conference46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021
Country/TerritoryEstonia
CityTallinn
Period23/08/2127/08/21

Keywords

  • Compression
  • Constraint satisfaction problems
  • Edge-weighted clique
  • Kernelization

ASJC Scopus subject areas

  • Software

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