142,206 research outputs found
Kernelization of Vertex Cover by Structural Parameters
In the NP-complete problem Vertex Cover, one is given a graph G and an integer k and are asked whether there exists a vertex set S ⊆ V (G) with size at most k such that every edge of the graph is incident to a vertex in S. In this thesis we explore techniques to solve Vertex Cover using parameterized algorithms, with a particular focus on kernelization by structural parameters. We present two new polynomial kernels for Vertex Cover, one parameterized by the size of a minimum degree-2 modulator, and one parameterized by the size of a minimum pseudoforest modulator. A degree-2 modulator is a vertex set X ⊆ V (G) such that G-X has maximum degree two, and a pseudoforest modulator is a vertex set X ⊆ V (G) such that every connected component of G-X has at most one cycle. Specifically, we provide polynomial time algorithms that for an input graph G and an integer k, outputs a graph G' and an integer k' such that G has a vertex cover of size k if and only if G' has a vertex cover of size k'. Moreover, the number of vertices of G' is bounded by O(|X|^7) where |X| is the size of a minimum degree-2 modulator for G, or bounded by O(|X|^12) where |X| is the size a minimum pseudoforest modulator for G. Our result extends known results on structural kernelization for Vertex Cover.Master i InformatikkMAMN-INFINF39
Faster exponential-time algorithms in graphs of bounded average degree
We first show that the Traveling Salesman Problem in an n-vertex graph with
average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and
exponential space for a constant \eps_d depending only on d, where the
O*-notation suppresses factors polynomial in the input size. Thus, we
generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of
bounded degree.
Then, we move to the problem of counting perfect matchings in a graph. We
first present a simple algorithm for counting perfect matchings in an n-vertex
graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the
complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on
inclusion-exclusion principle instead of algebraic transformations. Building
upon this result, we show that the number of perfect matchings in an n-vertex
graph with average degree bounded by d can be computed in
O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the
constant obtained by us for the Traveling Salesman Problem in graphs of average
degree at most 2d.
Moreover we obtain a simple algorithm that counts the number of perfect
matchings in an n-vertex bipartite graph of average degree at most d in
O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of
Izumi and Wadayama [FOCS 2012].Comment: 10 page
Lossy Kernelization
In this paper we propose a new framework for analyzing the performance of
preprocessing algorithms. Our framework builds on the notion of kernelization
from parameterized complexity. However, as opposed to the original notion of
kernelization, our definitions combine well with approximation algorithms and
heuristics. The key new definition is that of a polynomial size
-approximate kernel. Loosely speaking, a polynomial size
-approximate kernel is a polynomial time pre-processing algorithm that
takes as input an instance to a parameterized problem, and outputs
another instance to the same problem, such that . Additionally, for every , a -approximate solution
to the pre-processed instance can be turned in polynomial time into a
-approximate solution to the original instance .
Our main technical contribution are -approximate kernels of
polynomial size for three problems, namely Connected Vertex Cover, Disjoint
Cycle Packing and Disjoint Factors. These problems are known not to admit any
polynomial size kernels unless . Our approximate
kernels simultaneously beat both the lower bounds on the (normal) kernel size,
and the hardness of approximation lower bounds for all three problems. On the
negative side we prove that Longest Path parameterized by the length of the
path and Set Cover parameterized by the universe size do not admit even an
-approximate kernel of polynomial size, for any , unless
. In order to prove this lower bound we need to combine
in a non-trivial way the techniques used for showing kernelization lower bounds
with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and
approximate kernel lower bounds for Set Cover and Hitting Set parameterized
by universe siz
Approximating the Minimum Equivalent Digraph
The MEG (minimum equivalent graph) problem is, given a directed graph, to
find a small subset of the edges that maintains all reachability relations
between nodes. The problem is NP-hard. This paper gives an approximation
algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its
analysis are based on the simple idea of contracting long cycles. (This result
is strengthened slightly in ``On strongly connected digraphs with bounded cycle
length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local
improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms
(1994
A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection
Given an undirected graph with edge costs and node weights, the minimum
bisection problem asks for a partition of the nodes into two parts of equal
weight such that the sum of edge costs between the parts is minimized. We give
a polynomial time bicriteria approximation scheme for bisection on planar
graphs.
Specifically, let be the total weight of all nodes in a planar graph .
For any constant , our algorithm outputs a bipartition of the
nodes such that each part weighs at most and the total cost
of edges crossing the partition is at most times the total
cost of the optimal bisection. The previously best known approximation for
planar minimum bisection, even with unit node weights, was . Our
algorithm actually solves a more general problem where the input may include a
target weight for the smaller side of the bipartition.Comment: To appear in STOC 201
Phase Transition in Matched Formulas and a Heuristic for Biclique Satisfiability
A matched formula is a CNF formula whose incidence graph admits a matching
which matches a distinct variable to every clause. We study phase transition in
a context of matched formulas and their generalization of biclique satisfiable
formulas. We have performed experiments to find a phase transition of property
"being matched" with respect to the ratio where is the number of
clauses and is the number of variables of the input formula . We
compare the results of experiments to a theoretical lower bound which was shown
by Franco and Gelder (2003). Any matched formula is satisfiable, moreover, it
remains satisfiable even if we change polarities of any literal occurrences.
Szeider (2005) generalized matched formulas into two classes having the same
property -- var-satisfiable and biclique satisfiable formulas. A formula is
biclique satisfiable if its incidence graph admits covering by pairwise
disjoint bounded bicliques. Recognizing if a formula is biclique satisfiable is
NP-complete. In this paper we describe a heuristic algorithm for recognizing
whether a formula is biclique satisfiable and we evaluate it by experiments on
random formulas. We also describe an encoding of the problem of checking
whether a formula is biclique satisfiable into SAT and we use it to evaluate
the performance of our heuristicComment: Conference version submitted to SOFSEM 2018
(https://beda.dcs.fmph.uniba.sk/sofsem2019/) 18 pages(17 without refernces),
3 figures, 8 tables, an algorithm pseudocod
Between Treewidth and Clique-width
Many hard graph problems can be solved efficiently when restricted to graphs
of bounded treewidth, and more generally to graphs of bounded clique-width. But
there is a price to be paid for this generality, exemplified by the four
problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that
are all FPT parameterized by treewidth but none of which can be FPT
parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7,
8]. We therefore seek a structural graph parameter that shares some of the
generality of clique-width without paying this price. Based on splits, branch
decompositions and the work of Vatshelle [18] on Maximum Matching-width, we
consider the graph parameter sm-width which lies between treewidth and
clique-width. Some graph classes of unbounded treewidth, like
distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph
Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized
by sm-width
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