Björn Schumacher | Searcharxiv

arXiv2025

Hardness of SetCover Reoptimization

We study hardness of reoptimization of the fundamental and hard to approximate SetCover problem. Reoptimization considers an instance together with a solution and a modified instance where the goal is to approximate the modified instance while utilizing the information gained by solution to the related instance. We study four different types of reoptimization for (weighted) SetCover: adding a set, removing a set, adding an element to the universe, and removing an element from the universe. A few of these cases are known to be easier to approximate than the classic SetCover problem. We show that all the other cases are essentially as hard to approximate as SetCover. The reoptimization problem of adding and removing an element in the unweighted case is known to admit a PTAS. For these settings we show that there is no EPTAS under common hardness assumptions via a novel combination of the classic way to show that a reoptimization problem is NP-hard and the relation between EPTAS and FPT.

cs.CC↗

arXiv2025

Convolution and Knapsack in Higher Dimensions

In the Knapsack problem, one is given the task of packing a knapsack of a given size with items in order to gain a packing with a high profit value. An important connection to the $(\max,+)$-convolution problem has been established, where knapsack solutions can be combined by building the convolution of two sequences. This observation has been used in recent years to give conditional lower bounds but also parameterized algorithms. In this paper we carry these results into higher dimensions. We consider Knapsack where items are characterized by multiple properties -- given through a vector -- and a knapsack that has a capacity vector. The packing must not exceed any of the given capacity constraints. In order to show a similar sub-quadratic lower bound we consider a multidimensional version of $(\max, +)$-convolution. We then consider variants of this problem introduced by Cygan et al. and prove that they are all equivalent in terms of algorithms that allow for a running time sub-quadratic in the number of entries of the array. We develop a parameterized algorithm to solve higher dimensional Knapsack. The techniques we apply are inspired by an algorithm introduced by Axiotis and Tzamos. We will show that even for higher dimensional Knapsack, we can reduce the problem to convolution on one-dimensional, concave sequences, leading to an $\mathcal{O}(dn + dD \cdot \max\{Π_{i=1}^d{t_i}, t_{\max}\log t_{\max}\})$ algorithm, where $D$ is the number of different weight vectors, $t$ the capacity vector and $d$ is the dimension of the problem. Then, we use the techniques to improve the approach of Eisenbrand and Weismantel to obtain an algorithm for Integer Linear Programming with upper bounds with running time $\mathcal{O}(dn) + D \cdot \mathcal{O}(d Δ)^{d(d+1)} + T_{\mathrm{LP}}$.

cs.DS↗