In the classic maximum coverage problem, we are given subsets T 1 , . . . , T m of a universe [n] along with an integer k and the objective is to find a subset S ⊆ [m] of size k that maximizes C(S) := | ∪ i∈S T i |. It is well-known that the greedy algorithm for this problem achieves an approximation ratio of (1 − e −1 ) and there is a matching inapproximability result. We note that in the maximum coverage problem if an element e ∈ [n] is covered by several sets, it is still counted only once. By contrast, if we change the problem and count each element e as many times as it is covered, then we obtain a linear objective function, C (∞) (S) = i∈S |T i |, which can be easily maximized under a cardinality constraint. We study the maximum -multi-coverage problem which naturally interpolates between these two extremes. In this problem, an element can be counted up to times but no more; hence, we consider maximizing the function C ( ) (S) = e∈ [n] min{ , |{i ∈ S : e ∈ T i }|}, subject to the constraint |S| ≤ k. Note that the case of = 1 corresponds to the standard maximum coverage setting and = ∞ gives us a linear objective. We develop an efficient approximation algorithm that achieves an approximation ratio of 1 − e − ! for the -multi-coverage problem. In particular, when = 2, this factor is 1 − 2e −2 ≈ 0.73 and as grows the approximation ratio behaves as 1 − 1 √ 2π. We also prove that this approximation

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.

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 . 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.

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