779 research outputs found
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 of items
together with a collection, , of distinct subsets of these items called
tests. We assume that is a test cover, i.e., for each pair of items
there is a test in containing exactly one of these items. The
objective is to find a minimum size subcollection of , which is still a
test cover. The generic parameterized version of {\sc Test Cover} is denoted by
-{\sc Test Cover}. Here, we are given and a
positive integer parameter as input and the objective is to decide whether
there is a test cover of size at most . We study four
parameterizations for {\sc Test Cover} and obtain the following:
(a) -{\sc Test Cover}, and -{\sc Test Cover} are fixed-parameter
tractable (FPT).
(b) -{\sc Test Cover} and -{\sc Test Cover} are
W[1]-hard. Thus, it is unlikely that these problems are FPT
Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems
In the {\sc Hitting Set} problem, we are given a collection of
subsets of a ground set and an integer , and asked whether has a
-element subset that intersects each set in . We consider two
parameterizations of {\sc Hitting Set} below tight upper bounds: and
. In both cases is the parameter. We prove that the first
parameterization is fixed-parameter tractable, but has no polynomial kernel
unless coNPNP/poly. The second parameterization is W[1]-complete,
but the introduction of an additional parameter, the degeneracy of the
hypergraph , makes the problem not only fixed-parameter
tractable, but also one with a linear kernel. Here the degeneracy of
is the minimum integer such that for each the
hypergraph with vertex set and edge set containing all edges of
without vertices in , has a vertex of degree at most
In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected
graph (a directed graph) on vertices and an integer , and asked
whether has a set of vertices such that for each vertex there is an edge (arc) from a vertex in to . {\sc Nonblocker} can be
viewed as a special case of {\sc Directed Nonblocker} (replace an undirected
graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that
{\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for
{\sc Directed Nonblocker}
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT
Hitting Set is a classic problem in combinatorial optimization. Its input
consists of a set system F over a finite universe U and an integer t; the
question is whether there is a set of t elements that intersects every set in
F. The Hitting Set problem parameterized by the size of the solution is a
well-known W[2]-complete problem in parameterized complexity theory. In this
paper we investigate the complexity of Hitting Set under various structural
parameterizations of the input. Our starting point is the folklore result that
Hitting Set is polynomial-time solvable if there is a tree T on vertex set U
such that the sets in F induce connected subtrees of T. We consider the case
that there is a treelike graph with vertex set U such that the sets in F induce
connected subgraphs; the parameter of the problem is a measure of how treelike
the graph is. Our main positive result is an algorithm that, given a graph G
with cyclomatic number k, a collection P of simple paths in G, and an integer
t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of
size t that hits all paths in P. It is based on a connection to the 2-SAT
problem in multiple valued logic. For other parameterizations we derive
W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic
Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was
corrected in this update.
Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning under structural
restrictions. All these problems involve two tasks: (i) identifying the
structure in the input as required by the restriction, and (ii) using the
identified structure to solve the reasoning task efficiently. We show that for
most of the considered problems, task (i) admits a polynomial-time
preprocessing to a problem kernel whose size is polynomial in a structural
problem parameter of the input, in contrast to task (ii) which does not admit
such a reduction to a problem kernel of polynomial size, subject to a
complexity theoretic assumption. As a notable exception we show that the
consistency problem for the AtMost-NValue constraint admits a polynomial kernel
consisting of a quadratic number of variables and domain values. Our results
provide a firm worst-case guarantees and theoretical boundaries for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541,
arXiv:1104.556
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