3,557 research outputs found
On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion
In this paper, we investigate the approximability of two node deletion
problems. Given a vertex weighted graph and a specified, or
"distinguished" vertex , MDD(min) is the problem of finding a minimum
weight vertex set such that becomes the
minimum degree vertex in ; and MDD(max) is the problem of
finding a minimum weight vertex set such that
becomes the maximum degree vertex in . These are known
-complete problems and have been studied from the parameterized complexity
point of view in previous work. Here, we prove that for any ,
both the problems cannot be approximated within a factor , unless . We also show that for any
, MDD(min) cannot be approximated within a factor on bipartite graphs, unless , and that for any , MDD(max) cannot be approximated within a
factor on bipartite graphs, unless . We give an factor approximation algorithm
for MDD(max) on general graphs, provided the degree of is . We
then show that if the degree of is , a similar result holds
for MDD(min). We prove that MDD(max) is -complete on 3-regular unweighted
graphs and provide an approximation algorithm with ratio when is a
3-regular unweighted graph. In addition, we show that MDD(min) can be solved in
polynomial time when is a regular graph of constant degree.Comment: 16 pages, 4 figures, submitted to Elsevier's Journal of Discrete
Algorithm
Covering problems in edge- and node-weighted graphs
This paper discusses the graph covering problem in which a set of edges in an
edge- and node-weighted graph is chosen to satisfy some covering constraints
while minimizing the sum of the weights. In this problem, because of the large
integrality gap of a natural linear programming (LP) relaxation, LP rounding
algorithms based on the relaxation yield poor performance. Here we propose a
stronger LP relaxation for the graph covering problem. The proposed relaxation
is applied to designing primal-dual algorithms for two fundamental graph
covering problems: the prize-collecting edge dominating set problem and the
multicut problem in trees. Our algorithms are an exact polynomial-time
algorithm for the former problem, and a 2-approximation algorithm for the
latter problem, respectively. These results match the currently known best
results for purely edge-weighted graphs.Comment: To appear in SWAT 201
Contraction blockers for graphs with forbidden induced paths.
We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pℓ-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs
On dynamic threshold graphs and related classes
This paper deals with the well known classes of threshold and difference graphs, both characterized by separators, i.e. node weight functions and thresholds. We design an efficient algorithm to find the minimum separator, and we show how to maintain minimum its value when the input (threshold or difference) graph is fully dynamic, i.e. edges/nodes are inserted/removed. Moreover, exploiting the data structure used for maintaining the minimality of the separator, we study the disjoint union and the join of two threshold graphs, showing that the resulting graphs are threshold signed graphs, i.e. a superclass of both threshold and difference graphs. Finally, we consider the complement operation on all the three introduced classes of graphs.
All these operations produce in output the modified graph in terms of their separator and require time linear w.r.t. the number of different degrees. We observe that recomputing from scratch the separator would run either in linear (for threshold and difference graphs) or quadratic (for threshold signed graphs) time w.r.t. the number of nodes of the graph
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