TY - GEN

T1 - On the Hardness of Compressing Weights

AU - Jansen, Bart M.P.

AU - Roy, Shivesh K.

AU - Włodarczyk, Micha

N1 - Publisher Copyright:
© Bart M. P. Jansen, Shivesh K. Roy, and Michał Włodarczyk; licensed under Creative Commons License CC-BY 4.0 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021).

PY - 2021/8/1

Y1 - 2021/8/1

N2 - 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.

AB - 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.

KW - Compression

KW - Constraint satisfaction problems

KW - Edge-weighted clique

KW - Kernelization

UR - http://www.scopus.com/inward/record.url?scp=85115368639&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2021.64

DO - 10.4230/LIPIcs.MFCS.2021.64

M3 - Conference contribution

AN - SCOPUS:85115368639

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021

A2 - Bonchi, Filippo

A2 - Puglisi, Simon J.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021

Y2 - 23 August 2021 through 27 August 2021

ER -