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

Abstract

In this article we focus on the parameterized complexity of theMultidimensional Binary Vector Assignment problem (called \BVA). An input ofthis problem is defined by $m$ disjoint sets $V^1, V^2, \dots, V^m$, eachcomposed 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 vectorfrom each set $V^i$. To each $m$-tuple we associate a $p$ dimensional vector byapplying the bit-wise AND operation on the $m$ vectors of the tuple. Theobjective is to minimize the total number of zeros in these $n$ vectors. mBVAcan be seen as a variant of multidimensional matching where hyperedges areimplicitly locally encoded via labels attached to vertices, but was originallyintroduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results ($ETH$-basedresults, $W$[2]-hardness and a kernel lower bound) according to severalparameters: the standard parameter $k$ i.e. the total number of zeros), as wellas two parameters above some guaranteed values.Comment: 16 pages, 6 figure