Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations

TL;DR

This paper investigates the parameterized complexity of the Multidimensional Binary Vector Assignment problem, providing fixed-parameter tractable algorithms and hardness results based on various parameters, including the total zeros and above-guarantee measures.

Contribution

It introduces FPT algorithms and complexity bounds for the extit{mBVA} problem under multiple parameterizations, extending understanding of its computational complexity.

Findings

FPT algorithms for standard and above-guarantee parameters

W[2]-hardness results based on ETH

Kernel lower bounds for certain parameters

Abstract

In this article we focus on the parameterized complexity of the Multidimensional Binary Vector Assignment problem (called \BVA). An input of this problem is defined by $m$ disjoint sets $V^1, V^2, \dots, V^m$, each composed of $n$ binary vectors of size $p$. An output is a set of $n$ disjoint $m$-tuples of vectors, where each $m$-tuple is obtained by picking one vector from each set $V^i$. To each $m$-tuple we associate a $p$ dimensional vector by applying the bit-wise AND operation on the $m$ vectors of the tuple. The objective is to minimize the total number of zeros in these $n$ vectors. mBVA can be seen as a variant of multidimensional matching where hyperedges are implicitly locally encoded via labels attached to vertices, but was originally introduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results ($ETH$-based results, $W$[2]-hardness and a kernel lower bound) according to several parameters: the standard parameter $k$ i.e. the total number of zeros), as well as two parameters above some guaranteed values.