Parameterized Intractability of Even Set and Shortest Vector Problem

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W [1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP) , in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W [1]-hard to approximate (under randomized reductions) to some constant factor.

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Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

BRICS Report Series ◽

10.7146/brics.v11i19.21844 ◽

2004 ◽

Vol 11 (19) ◽

Cited By ~ 3

Author(s):

Bolette Ammitzbøll Madsen ◽

Peter Rossmanith

Keyword(s):

Time Complexity ◽

Exact Algorithm ◽

Conjunctive Normal Form ◽

Exact Algorithms ◽

Fixed Parameter Tractable ◽

Complete Problem ◽

Satisfiability Problems ◽

Fixed Parameter ◽

Np Complete ◽

Open Question

Inspired by the Maximum Satisfiability and Exact Satisfiability problems we present two Maximum Exact Satisfiability problems. The first problem called Maximum Exact Satisfiability is: given a formula in conjunctive normal form and an integer k, is there an assignment to all variables in the formula such that at least k clauses have exactly one true literal. The second problem called Restricted Maximum Exact Satisfiability has the further restriction that no clause is allowed to have more than one true literal. Both problems are proved NP-complete restricted to the versions where each clause contains at most two literals. In fact Maximum Exact Satisfiability is a generalisation of the well-known NP-complete problem MaxCut. We present an exact algorithm for Maximum Exact Satisfiability where each clause contains at most two literals with time complexity O(poly(L) . 2^{m/4}), where m is the number of clauses and L is the length of the formula. For the second version we give an algorithm with time complexity O(poly(L) . 1.324718^n) , where n is the number of variables. We note that when restricted to the versions where each clause contains exactly two literals and there are no negations both problems are fixed parameter tractable. It is an open question if this is also the case for the general problems.

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Space-efficient classical and quantum algorithms for the shortest vector problem

Quantum Information and Computation ◽

10.26421/qic18.3-4-7 ◽

2018 ◽

Vol 18 (3&4) ◽

pp. 283-305

Author(s):

Yanlin Chen ◽

Kai-Min Chung ◽

Ching-Yi Lai

Keyword(s):

Efficient Algorithm ◽

Coding Theory ◽

Quantum Computer ◽

Quantum Algorithms ◽

Nonzero Vector ◽

Vector Problem ◽

Shortest Vector Problem ◽

Linearly Independent ◽

Classical Algorithm ◽

Quantum Version

A lattice is the integer span of some linearly independent vectors. Lattice problems have many significant applications in coding theory and cryptographic systems for their conjectured hardness. The Shortest Vector Problem (SVP), which asks to find a shortest nonzero vector in a lattice, is one of the well-known problems that are believed to be hard to solve, even with a quantum computer. In this paper we propose space-efficient classical and quantum algorithms for solving SVP. Currently the best time-efficient algorithm for solving SVP takes 2^{n+o(n)} time and 2^{n+o(n)} space. Our classical algorithm takes 2^{2.05n+o(n)} time to solve SVP and it requires only 2^{0.5n+o(n)} space. We then adapt our classical algorithm to a quantum version, which can solve SVP in time 2^{1.2553n+o(n)} with 2^{0.5n+o(n)} classical space and only poly(n) qubits.

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