8,593 research outputs found
Coalition structure generation over graphs
We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) â R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members â that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k â„ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph
Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
The conventional theory of solids is well suited to describing band
structures locally near isolated points in momentum space, but struggles to
capture the full, global picture necessary for understanding topological
phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298
(2017)], we have introduced the way to overcome this difficulty by formulating
the problem of sewing together many disconnected local "k-dot-p" band
structures across the Brillouin zone in terms of graph theory. In the current
manuscript we give the details of our full theoretical construction. We show
that crystal symmetries strongly constrain the allowed connectivities of energy
bands, and we employ graph-theoretic techniques such as graph connectivity to
enumerate all the solutions to these constraints. The tools of graph theory
allow us to identify disconnected groups of bands in these solutions, and so
identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529
v2: Accepted version, minor typos corrected and references added. Now
19+epsilon page
Stability and bifurcations for dissipative polynomial automorphisms of C^2
We study stability and bifurcations in holomorphic families of polynomial
automorphisms of C^2. We say that such a family is weakly stable over some
parameter domain if periodic orbits do not bifurcate there. We first show that
this defines a meaningful notion of stability, which parallels in many ways the
classical notion of J-stability in one-dimensional dynamics. In the second part
of the paper, we prove that under an assumption of moderate dissipativity, the
parameters displaying homoclinic tangencies are dense in the bifurcation locus.
This confirms one of Palis' Conjectures in the complex setting. The proof
relies on the formalism of semi-parabolic bifurcation and the construction of
"critical points" in semi-parabolic basins (which makes use of the classical
Denjoy-Carleman-Ahlfors and Wiman Theorems).Comment: Revised version. Part 1 on holomorphic motions and stability was
reorganize
On the residual and profinite closures of commensurated subgroups
The residual closure of a subgroup of a group is the intersection of
all virtually normal subgroups of containing . We show that if is
generated by finitely many cosets of and if is commensurated, then the
residual closure of in is virtually normal. This implies that separable
commensurated subgroups of finitely generated groups are virtually normal. A
stream of applications to separable subgroups, polycyclic groups, residually
finite groups, groups acting on trees, lattices in products of trees and
just-infinite groups then flows from this main result.Comment: 22 page
One-dimensional disordered Ising models by replica and cavity methods
Using a formalism based on the spectral decomposition of the replicated
transfer matrix for disordered Ising models, we obtain several results that
apply both to isolated one-dimensional systems and to locally tree-like graph
and factor graph (p-spin) ensembles. We present exact analytical expressions,
which can be efficiently approximated numerically, for many types of
correlation functions and for the average free energies of open and closed
finite chains. All the results achieved, with the exception of those involving
closed chains, are then rigorously derived without replicas, using a
probabilistic approach with the same flavour of cavity method
Unavoidable topological minors of infinite graphs
This is the post-print version of the Article - Copyright @ 2010 ElsevierA graph G is loosely-c-connected, or â-c-connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of â-c-connected infinite graphs that form an unavoidable set among the topological minors of â-c-connected infinite graphs. Corresponding results for minors and parallel minors are also obtained.This study was supported in part by NSF grants DMS-1001230 and NSA grant H98230-10-1-018
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