Parameterized Intractability of Even Set and Shortest Vector Problem

The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over 𝔽_2, 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, or in other words, whether there is a nonzero vector 𝐱 such that 𝐀𝐱 has at most k nonzero coordinates. The question of whether k-Even Set is fixed parameter tractable (FPT) parameterized by the distance k has been repeatedly raised in 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 k-Even Set is W[1]-hard under randomized reductions. 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 the ℓ_p norm for some fixed p) is at most k. Similar to k-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 p > 1, k-SVP is W[1]-hard to approximate (under randomized reductions) to some constant factor.