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.

中文翻译：

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偶集和最短向量问题的参数化难解性

这 - 偶数集问题是最小距离问题线性码超过 ，可以表述如下：给定一个生成矩阵 和一个整数 ，判断生成的代码是否 最多有距离 ，或者换句话说，是否存在非零向量 这样 最多有 非零坐标。是否的问题 -Even Set 是由距离参数化的固定参数可处理 (FPT) 在文献中多次被提出；事实上，它是 Downey and Fellows [1999] 的开创性著作中为数不多的未解决问题之一。在这项工作中，我们表明 - 偶数集是W[1] - 随机减少下的困难。我们还考虑参数化 -最短向量问题（SVP），其中给定一个格子，其基向量是整数和整数 , 目标是确定最短向量的范数（在 一些固定的规范 ) 最多 . 相似 -Even Set，理解这个问题的复杂性也是Parameterized Complexity领域长期悬而未决的问题。我们证明，对于任何 , -SVP 是W[1]-难以近似（在随机减少下）到某个常数因子。