Parameterized Study of the Test Cover Problem

We carry out a systematic study of a natural covering problem, used for identification across several areas, in the realm of parameterized complexity. In the {sc Test Cover} problem we are given a set $[n]={1,...,n}$ of items together with a collection, $cal T$, of distinct subsets of these items called tests. We assume that $cal T$ is a test cover, i.e., for each pair of items there is a test in $cal T$ containing exactly one of these items. The objective is to find a minimum size subcollection of $cal T$, which is still a test cover. The generic parameterized version of {sc Test Cover} is denoted by $p(k,n,|{cal T}|)$-{sc Test Cover}. Here, we are given $([n],cal{T})$ and a positive integer parameter $k$ as input and the objective is to decide whether there is a test cover of size at most $p(k,n,|{cal T}|)$. We study four parameterizations for {sc Test Cover} and obtain the following: (a) $k$-{sc Test Cover}, and $(n-k)$-{sc Test Cover} are fixed-parameter tractable (FPT). (b) $(|{cal T}|-k)$-{sc Test Cover} and $(log n+k)$-{sc Test Cover} are W[1]-hard. Thus, it is unlikely that these problems are FPT.

Fixed-parameter tractability of satisfying beyond the number of variables

We consider a CNF formula $F$ as a multiset of clauses: $F={c_1,..., c_m}$. The set of variables of $F$ will be denoted by $V(F)$. Let $B_F$ denote the bipartite graph with partite sets $V(F)$ and $F$ and with an edge between $v in V(F)$ and $c in F$ if $v in c$ or $bar{v} in c$. The matching number $ u(F)$ of $F$ is the size of a maximum matching in $B_F$. In our main result, we prove that the following parameterization of {sc MaxSat} (denoted by $( u(F)+k)$-textsc{SAT}) is fixed-parameter tractable: Given a formula $F$, decide whether we can satisfy at least $ u(F)+k$ clauses in $F$, where $k$ is the parameter. A formula $F$ is called variable-matched if $ u(F)=|V(F)|.$ Let $delta(F)=|F|-|V(F)|$ and $delta^*(F)=max_{Fsubseteq F} delta(F).$ Our main result implies fixed-parameter tractability of {sc MaxSat} parameterized by $delta(F)$ for variable-matched formulas $F$; this complements related results of Kullmann (2000) and Szeider (2004) for {sc MaxSat} parameterized by $delta^*(F)$. To obtain our main result, we reduce $( u(F)+k)$-textsc{SAT} into the following parameterization of the {sc Hitting Set} problem (denoted by $(m-k)$-{sc Hitting Set}): given a collection $cal C$ of $m$ subsets of a ground set $U$ of $n$ elements, decide whether there is $Xsubseteq U$ such that $Ccap X eq emptyset$ for each $Cin cal C$ and $|X|le m-k,$ where $k$ is the parameter. Gutin, Jones and Yeo (2011) proved that $(m-k)$-{sc Hitting Set} is fixed-parameter tractable by obtaining an exponential kernel for the problem. We obtain two algorithms for $(m-k)$-{sc Hitting Set}: a deterministic algorithm of runtime $O((2e)^{2k+O(log^2 k)} (m+n)^{O(1)})$ and a randomized algorithm of expected runtime $O(8^{k+O(sqrt{k})} (m+n)^{O(1)})$. Our deterministic algorithm improves an algorithm that follows from the kernelization result of Gutin, Jones and Yeo (2011).

(Non-)existence of Polynomial Kernels for the Test Cover Problem

The input of the Test Cover problem consists of a set $V$ of vertices, and a collection ${cal E}={E_1,..., E_m}$ of distinct subsets of $V$, called tests. A test $E_q$ separates a pair $v_i,v_j$ of vertices if $|{v_i,v_j}cap E_q|=1.$ A subcollection ${cal T}subseteq {cal E}$ is a test cover if each pair $v_i,v_j$ of distinct vertices is separated by a test in ${cal T}$. The objective is to find a test cover of minimum cardinality, if one exists. This problem is NP-hard. We consider two parameterizations the Test Cover problem with parameter $k$: (a) decide whether there is a test cover with at most $k$ tests, (b) decide whether there is a test cover with at most $|V|-k$ tests. Both parameterizations are known to be fixed-parameter tractable. We prove that none have a polynomial size kernel unless $NPsubseteq coNP/poly$. Our proofs use the cross-composition method recently introduced by Bodlaender et al. (2011) and parametric duality introduced by Chen et al. (2005). The result for the parameterization (a) was an open problem (private communications with Henning Fernau and Jiong Guo, Jan.-Feb. 2012). We also show that the parameterization (a) admits a polynomial size kernel if the size of each test is upper-bounded by a constant.

Spanning directed trees with many leaves

The {sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that - every strongly connected $n$-vertex digraph $D$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; - if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph UG($D$) is $O(klog k)$. Moreover, if the digraph is acyclic, the pathwidth is at most $4k$. The last result implies that it can be decided in time $2^{O(klog^2 k)}cdot n^{O(1)}$ whether a strongly connected digraph on $n$ vertices has an out-branching with at least $k$ leaves. On acyclic digraphs the running time of our algorithm is $2^{O(klog k)}cdot n^{O(1)}$.