67,486 research outputs found
Parameterized complexity of DPLL search procedures
We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas
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
Simplified Energy Landscape for Modularity Using Total Variation
Networks capture pairwise interactions between entities and are frequently
used in applications such as social networks, food networks, and protein
interaction networks, to name a few. Communities, cohesive groups of nodes,
often form in these applications, and identifying them gives insight into the
overall organization of the network. One common quality function used to
identify community structure is modularity. In Hu et al. [SIAM J. App. Math.,
73(6), 2013], it was shown that modularity optimization is equivalent to
minimizing a particular nonconvex total variation (TV) based functional over a
discrete domain. They solve this problem, assuming the number of communities is
known, using a Merriman, Bence, Osher (MBO) scheme.
We show that modularity optimization is equivalent to minimizing a convex
TV-based functional over a discrete domain, again, assuming the number of
communities is known. Furthermore, we show that modularity has no convex
relaxation satisfying certain natural conditions. We therefore, find a
manageable non-convex approximation using a Ginzburg Landau functional, which
provably converges to the correct energy in the limit of a certain parameter.
We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et
al. and which is 7 times faster at solving the associated diffusion equation
due to the fact that the underlying discretization is unconditionally stable.
Our numerical tests include a hyperspectral video whose associated graph has
2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper
of Hu et al.Comment: 25 pages, 3 figures, 3 tables, submitted to SIAM J. App. Mat
Tree resolution proofs of the weak pigeon-hole principle.
We prove that any optimal tree resolution proof of PHPn m is of size 2&thetas;(n log n), independently from m, even if it is infinity. So far, only a 2Ω(n) lower bound has been known in the general case. We also show that any, not necessarily optimal, regular tree resolution proof PHPn m is bounded by 2O(n log m). To the best of our knowledge, this is the first time the worst case proof complexity has been considered. Finally, we discuss possible connections of our result to Riis' (1999) complexity gap theorem for tree resolution
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