The Satisfiability Threshold of Random 3-SAT Is at Least 3.52

We prove that a random 3-SAT instance with clause-to-variable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement

The Satisfiability Threshold for k-XORSAT

We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \geq 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability of constrained $k$-XORSAT for every $k\ge 3$, and we use standard results on the 2-core of a random $k$-uniform hypergraph to extend this result to find the threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow the phase transition window, showing that $m-n \to -\infty$ implies almost-sure satisfiability, while $m-n \to +\infty$ implies almost-sure unsatisfiability.Comment: Version 2 adds sharper phase transition result, new citation in literature survey, and improvements in presentation; removes Appendix treating k=

Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, notably Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for a more precise and improved running time analysi

Efficient algorithms for three-dimensional axial and planar random assignment problems

Beautiful formulas are known for the expected cost of random two-dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural “Axial” and “Planar” versions, both of which are NP-hard. For 3-dimensional Axial random assignment instances of size n, the cost scales as Ω(1/ n), and a main result of the present paper is a linear-time algorithm that, with high probability, finds a solution of cost O(n–1+o(1)). For 3-dimensional Planar assignment, the lower bound is Ω(n), and we give a new efficient matching-based algorithm that with high probability returns a solution with cost O(n log n)

The distribution of minimum-weight cliques and other subgraphs in graphs with random edge weights

We determine, asymptotically in n, the distribution and mean of the weight of a minimum-weight k-clique (or any strictly balanced graph H) in a complete graph Kn whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions. For the clique, we also provide explicit (non-asymptotic) bounds on the distribution's CDF in a form obtained directly from the Stein-Chen method, and in a looser but simpler form. The direct form extends to other subgraphs and other edge-weight distributions. We illustrate the clique results for various values of k and n. The results may be applied to evaluate whether an observed minimum-weight copy of a graph H in a network provides statistical evidence that the network's edge weights are not independently distributed but have some structure

Phase coexistence and torpid mixing in the 3-coloring model on Z^d

We show that for all sufficiently large d, the uniform proper 3-coloring model (in physics called the 3-state antiferromagnetic Potts model at zero temperature) on Z^d admits multiple maximal-entropy Gibbs measures. This is a consequence of the following combinatorial result: if a proper 3-coloring is chosen uniformly from a box in Z^d, conditioned on color 0 being given to all the vertices on the boundary of the box which are at an odd distance from a fixed vertex v in the box, then the probability that v gets color 0 is exponentially small in d. The proof proceeds through an analysis of a certain type of cutset separating v from the boundary of the box, and builds on techniques developed by Galvin and Kahn in their proof of phase transition in the hard-core model on Z^d. Building further on these techniques, we study local Markov chains for sampling proper 3-colorings of the discrete torus Z^d_n. We show that there is a constant \rho \approx 0.22 such that for all even n \geq 4 and d sufficiently large, if M is a Markov chain on the set of proper 3-colorings of Z^d_n that updates the color of at most \rho n^d vertices at each step and whose stationary distribution is uniform, then the mixing time of M (the time taken for M to reach a distribution that is close to uniform, starting from an arbitrary coloring) is essentially exponential in n^{d-1}

Stable non-uniform black strings below the critical dimension

The higher-dimensional vacuum Einstein equation admits translationally non-uniform black string solutions. It has been argued that infinitesimally non-uniform black strings should be unstable in 13 or fewer dimensions and otherwise stable. We construct numerically non-uniform black string solutions in 11, 12, 13, 14 and 15 dimensions. Their stability is investigated using local Penrose inequalities. Weakly non-uniform solutions behave as expected. However, in 12 and 13 dimensions, strongly non-uniform solutions appear to be stable and can have greater horizon area than a uniform string of the same mass. In 14 and 15 dimensions all non-uniform black strings appear to be stable.Comment: 26 pages, 11 figures. V2: reference added, matches published versio