184 research outputs found
Paradigms for Parameterized Enumeration
The aim of the paper is to examine the computational complexity and
algorithmics of enumeration, the task to output all solutions of a given
problem, from the point of view of parameterized complexity. First we define
formally different notions of efficient enumeration in the context of
parameterized complexity. Second we show how different algorithmic paradigms
can be used in order to get parameter-efficient enumeration algorithms in a
number of examples. These paradigms use well-known principles from the design
of parameterized decision as well as enumeration techniques, like for instance
kernelization and self-reducibility. The concept of kernelization, in
particular, leads to a characterization of fixed-parameter tractable
enumeration problems.Comment: Accepted for MFCS 2013; long version of the pape
A shortcut to (sun)flowers: Kernels in logarithmic space or linear time
We investigate whether kernelization results can be obtained if we restrict
kernelization algorithms to run in logarithmic space. This restriction for
kernelization is motivated by the question of what results are attainable for
preprocessing via simple and/or local reduction rules. We find kernelizations
for d-Hitting Set(k), d-Set Packing(k), Edge Dominating Set(k) and a number of
hitting and packing problems in graphs, each running in logspace. Additionally,
we return to the question of linear-time kernelization. For d-Hitting Set(k) a
linear-time kernelization was given by van Bevern [Algorithmica (2014)]. We
give a simpler procedure and save a large constant factor in the size bound.
Furthermore, we show that we can obtain a linear-time kernel for d-Set
Packing(k) as well.Comment: 18 page
A structural approach to kernels for ILPs: Treewidth and Total Unimodularity
Kernelization is a theoretical formalization of efficient preprocessing for
NP-hard problems. Empirically, preprocessing is highly successful in practice,
for example in state-of-the-art ILP-solvers like CPLEX. Motivated by this,
previous work studied the existence of kernelizations for ILP related problems,
e.g., for testing feasibility of Ax <= b. In contrast to the observed success
of CPLEX, however, the results were largely negative. Intuitively, practical
instances have far more useful structure than the worst-case instances used to
prove these lower bounds.
In the present paper, we study the effect that subsystems with (Gaifman graph
of) bounded treewidth or totally unimodularity have on the kernelizability of
the ILP feasibility problem. We show that, on the positive side, if these
subsystems have a small number of variables on which they interact with the
remaining instance, then we can efficiently replace them by smaller subsystems
of size polynomial in the domain without changing feasibility. Thus, if large
parts of an instance consist of such subsystems, then this yields a substantial
size reduction. We complement this by proving that relaxations to the
considered structures, e.g., larger boundaries of the subsystems, allow
worst-case lower bounds against kernelization. Thus, these relaxed structures
can be used to build instance families that cannot be efficiently reduced, by
any approach.Comment: Extended abstract in the Proceedings of the 23rd European Symposium
on Algorithms (ESA 2015
Constraint satisfaction parameterized by solution size
In the constraint satisfaction problem (CSP) corresponding to a constraint
language (i.e., a set of relations) , the goal is to find an assignment
of values to variables so that a given set of constraints specified by
relations from is satisfied. The complexity of this problem has
received substantial amount of attention in the past decade. In this paper we
study the fixed-parameter tractability of constraint satisfaction problems
parameterized by the size of the solution in the following sense: one of the
possible values, say 0, is "free," and the number of variables allowed to take
other, "expensive," values is restricted. A size constraint requires that
exactly variables take nonzero values. We also study a more refined version
of this restriction: a global cardinality constraint prescribes how many
variables have to be assigned each particular value. We study the parameterized
complexity of these types of CSPs where the parameter is the required number
of nonzero variables. As special cases, we can obtain natural and
well-studied parameterized problems such as Independent Set, Vertex Cover,
d-Hitting Set, Biclique, etc.
In the case of constraint languages closed under substitution of constants,
we give a complete characterization of the fixed-parameter tractable cases of
CSPs with size constraints, and we show that all the remaining problems are
W[1]-hard. For CSPs with cardinality constraints, we obtain a similar
classification, but for some of the problems we are only able to show that they
are Biclique-hard. The exact parameterized complexity of the Biclique problem
is a notorious open problem, although it is believed to be W[1]-hard.Comment: To appear in SICOMP. Conference version in ICALP 201
On the Computational Complexity of Vertex Integrity and Component Order Connectivity
The Weighted Vertex Integrity (wVI) problem takes as input an -vertex
graph , a weight function , and an integer . The
task is to decide if there exists a set such that the weight
of plus the weight of a heaviest component of is at most . Among
other results, we prove that:
(1) wVI is NP-complete on co-comparability graphs, even if each vertex has
weight ;
(2) wVI can be solved in time;
(3) wVI admits a kernel with at most vertices.
Result (1) refutes a conjecture by Ray and Deogun and answers an open
question by Ray et al. It also complements a result by Kratsch et al., stating
that the unweighted version of the problem can be solved in polynomial time on
co-comparability graphs of bounded dimension, provided that an intersection
model of the input graph is given as part of the input.
An instance of the Weighted Component Order Connectivity (wCOC) problem
consists of an -vertex graph , a weight function ,
and two integers and , and the task is to decide if there exists a set
such that the weight of is at most and the weight of
a heaviest component of is at most . In some sense, the wCOC problem
can be seen as a refined version of the wVI problem. We prove, among other
results, that:
(4) wCOC can be solved in time on interval graphs,
while the unweighted version can be solved in time on this graph
class;
(5) wCOC is W[1]-hard on split graphs when parameterized by or by ;
(6) wCOC can be solved in time;
(7) wCOC admits a kernel with at most vertices.
We also show that result (6) is essentially tight by proving that wCOC cannot
be solved in time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the
conference proceedings of ISAAC 201
Reduction Techniques for Graph Isomorphism in the Context of Width Parameters
We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure
On the stable degree of graphs
We define the stable degree s(G) of a graph G by s(G)∈=∈ min max d (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k∈≥∈3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree
Polynomial kernelization for removing induced claws and diamonds
A graph is called (claw,diamond)-free if it contains neither a claw (a
) nor a diamond (a with an edge removed) as an induced subgraph.
Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of
triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex
is in at most two maximal cliques and every edge is in exactly one maximal
clique.
In this paper we consider the parameterized complexity of the
(claw,diamond)-free Edge Deletion problem, where given a graph and a
parameter , the question is whether one can remove at most edges from
to obtain a (claw,diamond)-free graph. Our main result is that this problem
admits a polynomial kernel. We complement this finding by proving that, even on
instances with maximum degree , the problem is NP-complete and cannot be
solved in time unless the Exponential Time
Hypothesis fai
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