Extremal Optimization at the Phase Transition of the 3-Coloring Problem

We investigate the phase transition of the 3-coloring problem on random graphs, using the extremal optimization heuristic. 3-coloring is among the hardest combinatorial optimization problems and is closely related to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the ``backbone'', an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size $n$ up to 512. For graphs up to this size, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about $O(n^{3.5})$ update steps. Finite size scaling gives a critical mean degree value $\alpha_{\rm c}=4.703(28)$. Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available at http://www.physics.emory.edu/faculty/boettcher

Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure

Testing Gaugino Mass Unification Directly at the LHC

We report on the first step of a systematic study of how gaugino mass unification can be probed at the LHC in a quasi-model independent manner. Here we focus our attention on the theoretically well-motivated mirage pattern of gaugino masses, a one-parameter family of models of which universal (high scale) gaugino masses are a limiting case. Using a statistical method to optimize our signature selection we arrive at three ensembles of observables targeted at the physics of the gaugino sector, allowing for a determination of this non-universality parameter without reconstructing individual mass eigenvalues or the soft supersymmetry-breaking gaugino masses themselves. In this controlled environment we find that approximately 80% of the supersymmetric parameter space would give rise to a model for which our method will detect non-universality in the gaugino mass sector at the 10% level with approximately 10 inverse femptobarns of integrated luminosity.Comment: To appear in proceedings of "Beyond the Standard Model at the LHC (BSM-LHC)", June 2-4, 200

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat). The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem

Explicit Optimal Hardness via Gaussian stability results

The results of Raghavendra (2008) show that assuming Khot's Unique Games Conjecture (2002), for every constraint satisfaction problem there exists a generic semi-definite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate exactly the Unique Games hardness of the problem. Obtaining an explicit optimal approximation scheme and the corresponding approximation factor is a difficult challenge for each specific approximation problem. An approach for determining the exact approximation factor and the corresponding optimal rounding was established in the analysis of MAX-CUT (KKMO 2004) and the use of the Invariance Principle (MOO 2005). However, this approach crucially relies on results explicitly proving optimal partitions in Gaussian space. Until recently, Borell's result (Borell 1985) was the only non-trivial Gaussian partition result known. In this paper we derive the first explicit optimal approximation algorithm and the corresponding approximation factor using a new result on Gaussian partitions due to Isaksson and Mossel (2012). This Gaussian result allows us to determine exactly the Unique Games Hardness of MAX-3-EQUAL. In particular, our results show that Zwick algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is $\approx 0.796$ as conjectured by Zwick. We further use the previously known optimal Gaussian partitions results to obtain a new Unique Games Hardness factor for MAX-k-CSP : Using the well known fact that jointly normal pairwise independent random variables are fully independent, we show that the the UGC hardness of Max-k-CSP is $\frac{\lceil (k+1)/2 \rceil}{2^{k-1}}$, improving on results of Austrin and Mossel (2009)