12,023 research outputs found
On the Separation of Topology-Free Rank Inequalities for the Max Stable Set Problem
In the context of finding the largest stable set of a graph, rank inequalities prescribe that a stable set can contain, from any induced subgraph of the original graph, at most as many vertices as the stability number of the former. Although these inequalities subsume many of the valid inequalities known for the problem, their exact separation has only been investigated in few special cases obtained by restricting the induced subgraph to a specific topology. In this work, we propose a different approach in which, rather than imposing topological restrictions on the induced subgraph, we assume the right-hand side of the inequality to be fixed to a given (but arbitrary) constant. We then study the arising separation problem, which corresponds to the problem of finding a maximum weight subgraph with a bounded stability number. After proving its hardness and giving some insights on its polyhedral structure, we propose an exact branch-and-cut method for its solution. Computational results show that the separation of topology-free rank inequalities with a fixed right-hand side yields a substantial improvement over the bound provided by the fractional clique polytope (which is obtained with rank inequalities where the induced subgraph is restricted to a clique), often better than that obtained with Lovasz's Theta function via semidefmite programming
A polynomial algorithm for the k-cluster problem on interval graphs
This paper deals with the problem of finding, for a given graph and a given
natural number k, a subgraph of k nodes with a maximum number of edges. This
problem is known as the k-cluster problem and it is NP-hard on general graphs
as well as on chordal graphs. In this paper, it is shown that the k-cluster
problem is solvable in polynomial time on interval graphs. In particular, we
present two polynomial time algorithms for the class of proper interval graphs
and the class of general interval graphs, respectively. Both algorithms are
based on a matrix representation for interval graphs. In contrast to
representations used in most of the previous work, this matrix representation
does not make use of the maximal cliques in the investigated graph.Comment: 12 pages, 5 figure
Hedonic Games with Graph-restricted Communication
We study hedonic coalition formation games in which cooperation among the
players is restricted by a graph structure: a subset of players can form a
coalition if and only if they are connected in the given graph. We investigate
the complexity of finding stable outcomes in such games, for several notions of
stability. In particular, we provide an efficient algorithm that finds an
individually stable partition for an arbitrary hedonic game on an acyclic
graph. We also introduce a new stability concept -in-neighbor stability- which
is tailored for our setting. We show that the problem of finding an in-neighbor
stable outcome admits a polynomial-time algorithm if the underlying graph is a
path, but is NP-hard for arbitrary trees even for additively separable hedonic
games; for symmetric additively separable games we obtain a PLS-hardness
result
On Minimum Maximal Distance-k Matchings
We study the computational complexity of several problems connected with
finding a maximal distance- matching of minimum cardinality or minimum
weight in a given graph. We introduce the class of -equimatchable graphs
which is an edge analogue of -equipackable graphs. We prove that the
recognition of -equimatchable graphs is co-NP-complete for any fixed . We provide a simple characterization for the class of strongly chordal
graphs with equal -packing and -domination numbers. We also prove that
for any fixed integer the problem of finding a minimum weight
maximal distance- matching and the problem of finding a minimum weight
-independent dominating set cannot be approximated in polynomial
time in chordal graphs within a factor of unless
, where is a fixed constant (thereby
improving the NP-hardness result of Chang for the independent domination case).
Finally, we show the NP-hardness of the minimum maximal induced matching and
independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure
A Spinorial Formulation of the Maximum Clique Problem of a Graph
We present a new formulation of the maximum clique problem of a graph in
complex space. We start observing that the adjacency matrix A of a graph can
always be written in the form A = B B where B is a complex, symmetric matrix
formed by vectors of zero length (null vectors) and the maximum clique problem
can be transformed in a geometrical problem for these vectors. This problem, in
turn, is translated in spinorial language and we show that each graph uniquely
identifies a set of pure spinors, that is vectors of the endomorphism space of
Clifford algebras, and the maximum clique problem is formalized in this setting
so that, this much studied problem, may take advantage from recent progresses
of pure spinor geometry
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