51 research outputs found
Composing dynamic programming tree-decomposition-based algorithms
Given two integers and as well as graph classes
, the problems
,
, and
ask, given graph
as input, whether , , respectively can be partitioned
into sets such that, for each between and
, , , respectively. Moreover in , we request that the number of edges with
endpoints in different sets of the partition is bounded by . We show that if
there exist dynamic programming tree-decomposition-based algorithms for
recognizing the graph classes , for each , then we can
constructively create a dynamic programming tree-decomposition-based algorithms
for ,
, and
. We show that, in
some known cases, the obtained running times are comparable to those of the
best know algorithms
Contraction-Bidimensionality of Geometric Intersection Graphs
Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Gamma_k. A graph class G has the SQGC property if every graph G in G has treewidth O(bcg(G)c) for some 1 <= c < 2. The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects
Non-monotone target sets for threshold values restricted to , , and the vertex degree
We consider a non-monotone activation process
on a graph , where , for every positive integer , and is a threshold function. The set is a so-called non-monotone
target set for if there is some such that for every
. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8
(2011) 87-96] asked whether a target set of minimum order can be determined
efficiently if is a tree. We answer their question in the affirmative for
threshold functions satisfying for every
vertex~. For such restricted threshold functions, we give a characterization
of target sets that allows to show that the minimum target set problem remains
NP-hard for planar graphs of maximum degree but is efficiently solvable for
graphs of bounded treewidth
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