48 research outputs found
Finding Biclique Partitions of Co-Chordal Graphs
The biclique partition number of a graph is referred to as
the least number of complete bipartite (biclique) subgraphs that are required
to cover the edges of the graph exactly once. In this paper, we show that the
biclique partition number () of a co-chordal (complementary graph of
chordal) graph is less than the number of maximal cliques
() of its complementary graph: a chordal graph . We
first provide a general framework of the ``divide and conquer" heuristic of
finding minimum biclique partitions of co-chordal graphs based on clique trees.
Furthermore, a heuristic of complexity is proposed by
applying lexicographic breadth-first search to find structures called moplexes.
Either heuristic gives us a biclique partition of with size
. In addition, we prove that both of our heuristics can solve
the minimum biclique partition problem on exactly if its complement
is chordal and clique vertex irreducible. We also show that if is a split graph
On relaxations of the max -cut problem formulations
A tight continuous relaxation is a crucial factor in solving mixed integer
formulations of many NP-hard combinatorial optimization problems. The
(weighted) max -cut problem is a fundamental combinatorial optimization
problem with multiple notorious mixed integer optimization formulations. In
this paper, we explore four existing mixed integer optimization formulations of
the max -cut problem. Specifically, we show that the continuous relaxation
of a binary quadratic optimization formulation of the problem is: (i) stronger
than the continuous relaxation of two mixed integer linear optimization
formulations and (ii) at least as strong as the continuous relaxation of a
mixed integer semidefinite optimization formulation. We also conduct a set of
experiments on multiple sets of instances of the max -cut problem using
state-of-the-art solvers that empirically confirm the theoretical results in
item (i). Furthermore, these numerical results illustrate the advances in the
efficiency of global non-convex quadratic optimization solvers and more general
mixed integer nonlinear optimization solvers. As a result, these solvers
provide a promising option to solve combinatorial optimization problems. Our
codes and data are available on GitHub
Study of forward Z + jet production in pp collisions at √s=7 TeV
A measurement of the +jet production cross-section in collisions at a centre-of-mass energy TeV is presented. The analysis is based on an integrated luminosity of recorded by the LHCb experiment. Results are shown with two jet transverse momentum thresholds, 10 and 20 GeV, for both the overall cross-section within the fiducial volume, and for six differential cross-section measurements. The fiducial volume requires that both the jet and the muons from the Z boson decay are produced in the forward direction (). The results show good agreement with theoretical predictions at the second-order expansion in the coupling of the strong interaction.A measurement of the +jet production cross-section in collisions at a centre-of-mass energy TeV is presented. The analysis is based on an integrated luminosity of recorded by the LHCb experiment. Results are shown with two jet transverse momentum thresholds, 10 and 20 GeV, for both the overall cross-section within the fiducial volume, and for six differential cross-section measurements. The fiducial volume requires that both the jet and the muons from the Z boson decay are produced in the forward direction (). The results show good agreement with theoretical predictions at the second-order expansion in the coupling of the strong interaction