1,847 research outputs found
Faster Deterministic Algorithms for Packing, Matching and -Dominating Set Problems
In this paper, we devise three deterministic algorithms for solving the
-set -packing, -dimensional -matching, and -dominating set
problems in time , and ,
respectively. Although recently there has been remarkable progress on
randomized solutions to those problems, our bounds make good improvements on
the best known bounds for deterministic solutions to those problems.Comment: ISAAC13 Submission. arXiv admin note: substantial text overlap with
arXiv:1303.047
Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints
In parameterized complexity, it is a natural idea to consider different
generalizations of classic problems. Usually, such generalization are obtained
by introducing a "relaxation" variable, where the original problem corresponds
to setting this variable to a constant value. For instance, the problem of
packing sets of size at most into a given universe generalizes the Maximum
Matching problem, which is recovered by taking . Most often, the
complexity of the problem increases with the relaxation variable, but very
recently Abasi et al. have given a surprising example of a problem ---
-Simple -Path --- that can be solved by a randomized algorithm with
running time . That is, the complexity of the
problem decreases with . In this paper we pursue further the direction
sketched by Abasi et al. Our main contribution is a derandomization tool that
provides a deterministic counterpart of the main technical result of Abasi et
al.: the algorithm for -Monomial
Detection, which is the problem of finding a monomial of total degree and
individual degrees at most in a polynomial given as an arithmetic circuit.
Our technique works for a large class of circuits, and in particular it can be
used to derandomize the result of Abasi et al. for -Simple -Path. On our
way to this result we introduce the notion of representative sets for
multisets, which may be of independent interest. Finally, we give two more
examples of problems that were already studied in the literature, where the
same relaxation phenomenon happens. The first one is a natural relaxation of
the Set Packing problem, where we allow the packed sets to overlap at each
element at most times. The second one is Degree Bounded Spanning Tree,
where we seek for a spanning tree of the graph with a small maximum degree
Parameterization Above a Multiplicative Guarantee
Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ? g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ?>0, multiplicative parameterization above g(I)^(1+?) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth
Improved approximation for 3-dimensional matching via bounded pathwidth local search
One of the most natural optimization problems is the k-Set Packing problem,
where given a family of sets of size at most k one should select a maximum size
subfamily of pairwise disjoint sets. A special case of 3-Set Packing is the
well known 3-Dimensional Matching problem. Both problems belong to the Karp`s
list of 21 NP-complete problems. The best known polynomial time approximation
ratio for k-Set Packing is (k + eps)/2 and goes back to the work of Hurkens and
Schrijver [SIDMA`89], which gives (1.5 + eps)-approximation for 3-Dimensional
Matching. Those results are obtained by a simple local search algorithm, that
uses constant size swaps.
The main result of the paper is a new approach to local search for k-Set
Packing where only a special type of swaps is considered, which we call swaps
of bounded pathwidth. We show that for a fixed value of k one can search the
space of r-size swaps of constant pathwidth in c^r poly(|F|) time. Moreover we
present an analysis proving that a local search maximum with respect to O(log
|F|)-size swaps of constant pathwidth yields a polynomial time (k + 1 +
eps)/3-approximation algorithm, improving the best known approximation ratio
for k-Set Packing. In particular we improve the approximation ratio for
3-Dimensional Matching from 3/2 + eps to 4/3 + eps.Comment: To appear in proceedings of FOCS 201
A randomized polynomial kernel for Subset Feedback Vertex Set
The Subset Feedback Vertex Set problem generalizes the classical Feedback
Vertex Set problem and asks, for a given undirected graph , a set , and an integer , whether there exists a set of at most
vertices such that no cycle in contains a vertex of . It was
independently shown by Cygan et al. (ICALP '11, SIDMA '13) and Kawarabayashi
and Kobayashi (JCTB '12) that Subset Feedback Vertex Set is fixed-parameter
tractable for parameter . Cygan et al. asked whether the problem also admits
a polynomial kernelization.
We answer the question of Cygan et al. positively by giving a randomized
polynomial kernelization for the equivalent version where is a set of
edges. In a first step we show that Edge Subset Feedback Vertex Set has a
randomized polynomial kernel parameterized by with
vertices. For this we use the matroid-based tools of Kratsch and Wahlstr\"om
(FOCS '12) that for example were used to obtain a polynomial kernel for
-Multiway Cut. Next we present a preprocessing that reduces the given
instance to an equivalent instance where the size of
is bounded by . These two results lead to a polynomial kernel for
Subset Feedback Vertex Set with vertices
Parameterized algorithms of fundamental NP-hard problems: a survey
Parameterized computation theory has developed rapidly over the last two decades. In theoretical computer science, it has attracted considerable attention for its theoretical value and significant guidance in many practical applications. We give an overview on parameterized algorithms for some fundamental NP-hard problems, including MaxSAT, Maximum Internal Spanning Trees, Maximum Internal Out-Branching, Planar (Connected) Dominating Set, Feedback Vertex Set, Hyperplane Cover, Vertex Cover, Packing and Matching problems. All of these problems have been widely applied in various areas, such as Internet of Things, Wireless Sensor Networks, Artificial Intelligence, Bioinformatics, Big Data, and so on. In this paper, we are focused on the algorithms’ main idea and algorithmic techniques, and omit the details of them
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
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