Parameterized intractability of even set and shortest vector problem from gap-ETH

Bhattacharyya, A and Ghoshal, S and Karthik, CS and Manurangsi, P (2018) Parameterized intractability of even set and shortest vector problem from gap-ETH. In: 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, 9 - 13 July 2018, Prague.

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Abstract

The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over F2, which can be stated as follows: given a generator matrix A and an integer k, determine whether the code generated by A has distance at most k. Here, k is the parameter of the problem. The question of whether k-Even Set is fixed parameter tractable (FPT) has been repeatedly raised in literature and has earned its place in Downey and Fellows' book (2013) as one of the “most infamous” open problems in the field of Parameterized Complexity. In this work, we show that k-Even Set does not admit FPT algorithms under the (randomized) Gap Exponential Time Hypothesis (Gap-ETH) [Dinur'16, Manurangsi-Raghavendra'16]. In fact, our result rules out not only exact FPT algorithms, but also any constant factor FPT approximation algorithms for the problem. Furthermore, our result holds even under the following weaker assumption, which is also known as the Parameterized Inapproximability Hypothesis (PIH) [Lokshtanov et al.'17]: no (randomized) FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only 0.99-satisfiable (where the parameter is the number of variables). We also consider the parameterized k-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer k, and the goal is to determine whether the norm of the shortest vector (in thep norm for some fixed p) is at most k. Similar to k-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity. We show that, for any p > 1, k-SVP is hard to approximate (in FPT time) to some constant factor, assuming PIH. Furthermore, for the case of p = 2, the inapproximability factor can be amplified to any constant.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing.
Keywords:	Approximation algorithms; Automata theory; Vectors, Even Set; Gap-ETH; Inapproximability; Minimum distance problems; Parameterized complexity; Shortest vector problem, Parameterization
Department/Centre:	Division of Chemical Sciences > Inorganic & Physical Chemistry
Date Deposited:	08 Aug 2022 10:30
Last Modified:	08 Aug 2022 10:30
URI:	https://eprints.iisc.ac.in/id/eprint/75491

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