Parameterized Intractability of Even Set and Shortest Vector Problem

Abstract

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.