4,458 research outputs found
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
Computing maximum cliques in -EPG graphs
EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection
graphs of paths on an orthogonal grid. The class -EPG is the subclass of
EPG graphs where the path on the grid associated to each vertex has at most
bends. Epstein et al. showed in 2013 that computing a maximum clique in
-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the
number of bends is at least , the class contains -interval graphs for
which computing a maximum clique is an NP-hard problem. The complexity status
of the Maximum Clique problem remains open for and -EPG graphs. In
this paper, we show that we can compute a maximum clique in polynomial time in
-EPG graphs given a representation of the graph.
Moreover, we show that a simple counting argument provides a
-approximation for the coloring problem on -EPG graphs without
knowing the representation of the graph. It generalizes a result of [Epstein et
al, 2013] on -EPG graphs (where the representation was needed)
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails
Solving Maximum Clique Problem for Protein Structure Similarity
A basic assumption of molecular biology is that proteins sharing close
three-dimensional (3D) structures are likely to share a common function and in
most cases derive from a same ancestor. Computing the similarity between two
protein structures is therefore a crucial task and has been extensively
investigated. Evaluating the similarity of two proteins can be done by finding
an optimal one-to-one matching between their components, which is equivalent to
identifying a maximum weighted clique in a specific "alignment graph". In this
paper we present a new integer programming formulation for solving such clique
problems. The model has been implemented using the ILOG CPLEX Callable Library.
In addition, we designed a dedicated branch and bound algorithm for solving the
maximum cardinality clique problem. Both approaches have been integrated in
VAST (Vector Alignment Search Tool) - a software for aligning protein 3D
structures largely used in NCBI (National Center for Biotechnology
Information). The original VAST clique solver uses the well known Bron and
Kerbosh algorithm (BK). Our computational results on real life protein
alignment instances show that our branch and bound algorithm is up to 116 times
faster than BK for the largest proteins
Exact Algorithms for Maximum Clique: a computational study
We investigate a number of recently reported exact algorithms for the maximum
clique problem (MCQ, MCR, MCS, BBMC). The program code used is presented and
critiqued showing how small changes in implementation can have a drastic effect
on performance. The computational study demonstrates how problem features and
hardware platforms influence algorithm behaviour. The minimum width order
(smallest-last) is investigated, and MCS is broken into its consituent parts
and we discover that one of these parts degrades performance. It is shown that
the standard procedure used for rescaling published results is unsafe.Comment: 40 pages, 14 figures, 10 tables, 12 short java program listings, code
afailable to download at
http://www.dcs.gla.ac.uk/~pat/maxClique/distribution
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