Optimization bounds from the branching dual

We present a general method for obtaining strong bounds for discrete optimization problems that is based on a concept of branching duality. It can be applied when no useful integer programming model is available, and we illustrate this with the minimum bandwidth problem. The method strengthens a known bound for a given problem by formulating a dual problem whose feasible solutions are partial branching trees. It solves the dual problem with a “worst-bound” local search heuristic that explores neighboring partial trees. After proving some optimality properties of the heuristic, we show that it substantially improves known combinatorial bounds for the minimum bandwidth problem with a modest amount of computation. It also obtains significantly tighter bounds than depth-first and breadth-first branching, demonstrating that the dual perspective can lead to better branching strategies when the object is to find valid bounds.Accepted manuscrip

Next nearest neighbour Ising models on random graphs

This paper develops results for the next nearest neighbour Ising model on random graphs. Besides being an essential ingredient in classic models for frustrated systems, second neighbour interactions interactions arise naturally in several applications such as the colour diversity problem and graphical games. We demonstrate ensembles of random graphs, including regular connectivity graphs, that have a periodic variation of free energy, with either the ratio of nearest to next nearest couplings, or the mean number of nearest neighbours. When the coupling ratio is integer paramagnetic phases can be found at zero temperature. This is shown to be related to the locked or unlocked nature of the interactions. For anti-ferromagnetic couplings, spin glass phases are demonstrated at low temperature. The interaction structure is formulated as a factor graph, the solution on a tree is developed. The replica symmetric and energetic one-step replica symmetry breaking solution is developed using the cavity method. We calculate within these frameworks the phase diagram and demonstrate the existence of dynamical transitions at zero temperature for cases of anti-ferromagnetic coupling on regular and inhomogeneous random graphs.Comment: 55 pages, 15 figures, version 2 with minor revisions, to be published J. Stat. Mec

Halo-independent tests of dark matter direct detection signals: local DM density, LHC, and thermal freeze-out

From an assumed signal in a Dark Matter (DM) direct detection experiment a lower bound on the product of the DM--nucleon scattering cross section and the local DM density is derived, which is independent of the local DM velocity distribution. This can be combined with astrophysical determinations of the local DM density. Within a given particle physics model the bound also allows a robust comparison of a direct detection signal with limits from the LHC. Furthermore, the bound can be used to formulate a condition which has to be fulfilled if the particle responsible for the direct detection signal is a thermal relic, regardless of whether it constitutes all DM or only part of it. We illustrate the arguments by adopting a simplified DM model with a Z' mediator and assuming a signal in a future xenon direct detection experiment.Comment: 23 pages, 6 figure

Finite Domain Bounds Consistency Revisited

A widely adopted approach to solving constraint satisfaction problems combines systematic tree search with constraint propagation for pruning the search space. Constraint propagation is performed by propagators implementing a certain notion of consistency. Bounds consistency is the method of choice for building propagators for arithmetic constraints and several global constraints in the finite integer domain. However, there has been some confusion in the definition of bounds consistency. In this paper we clarify the differences and similarities among the three commonly used notions of bounds consistency.Comment: 12 page