We analyse the performance of Belief Propagation Guided Decimation, a physics-inspired message passing algorithm, on the random $k$-XORSAT problem. Specifically, we derive an explicit threshold up to which the algorithm succeeds with a strictly positive probability $Ω(1)$ that we compute explicitly, but beyond which the algorithm with high probability fails to find a satisfying assignment. In addition, we analyse a thought experiment called the decimation process for which we identify a (non-) reconstruction and a condensation phase transition. The main results of the present work confirm physics predictions from [RTS: J. Stat. Mech. 2009] that link the phase transitions of the decimation process with the performance of the algorithm, and improve over partial results from a recent article [Yung: Proc. ICALP 2024].

The Ising antiferromagnet is an important statistical physics model with close connections to the Max Cut problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the Max Cut of random regular graphs predicted by Zdeborová and Boettcher [ J. Stat. Mech., 2010 (2010), P02020]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten--Stigum bound.

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 $K_n$ whose edge weights are independent random values drawn from the uniform distribution or other continuous distributions. For the clique, we also provide explicit (nonasymptotic) bounds on the distribution's cumulative distribution function 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.

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 analysis. We illustrate the method with improved algorithms for M ax ( r ,2)-C sp and #D ominating S et . An instance of the problem M ax ( r ,2)-C SP , or simply M ax 2-CSP, is parameterized by the domain size r (often 2), the number of variables n (vertices in the constraint graph G ), and the number of constraints m (edges in G ). When G is cubic, and omitting sub-exponential terms here for clarity, we give an algorithm running in time r (1/5) n = r (2/15) m the previous best was r (1/4) n = r (1/6) m . By known results, this improvement for the cubic case results in an algorithm running in time r (9/50) m for general instances; the previous best was r (19/100) m . We show that the analysis of the earlier algorithm was tight: our improvement is in the algorithm, not just the analysis. The same running time improvements hold for M ax C ut , an important special case of M ax 2-CSP, and for Polynomial and Ring CSP, generalizations encompassing graph bisection, the Ising model, and counting. We also give faster algorithms for #D ominating S et , counting the dominating sets of every cardinality 0, … , n for a graph G of order n . For cubic graphs, our algorithm runs in time 3 (1/5) n the previous best was 2 (1/2) n . For general graphs, we give an unrelated algorithm running in time 1.5183 n the previous best was 1.5673 n . The previous best algorithms for these problems all used local transformations and were analyzed by the Measure and Conquer method. Our new algorithms capitalize on the existence of small balanced separators for cubic graphs—a non-local property—and the ability to tailor the local algorithms always to “pivot” on a vertex in the separator. The new algorithms perform much as the old ones until the separator is empty, at which point they gain because the remaining vertices are split into two independent problem instances that can be solved recursively. It is likely that such algorithms can be effective for other problems too, and we present their design and analysis in a general framework.

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 ${\mathbb Z}^d$ 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 ${\mathbb Z}^d$, 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 ${\mathbb Z}^d$. Building further on these techniques, we study local Markov chains for sampling proper 3-colorings of the discrete torus ${\mathbb Z}^d_n$. We show that there is a constant $\rho \approx 0.22$ such that for all even $n \geq 4$ and $d$ sufficiently large, if ${\mathcal M}$ is a Markov chain on the set of proper 3-colorings of ${\mathbb Z}^d_n$ 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 ${\mathcal M}$ (the time taken for ${\mathcal M}$ to reach a distribution that is close to uniform, starting from an arbitrary coloring) is essentially exponential in $n^{d-1}$.

We introduce “hybrid” Max 2-CSP formulas consisting of “simple clauses”, namely conjunctions and disjunctions of pairs of variables, and general 2-variable clauses, which can be any integer-valued functions of pairs of boolean variables. This allows an algorithm to use both efficient reductions specific to AND and OR clauses, and other powerful reductions that require the general CSP setting. Parametrizing an instance by the fraction p of non-simple clauses, we give an exact (exponential-time) algorithm that is the fastest polynomial-space algorithm known for Max 2-Sat (and other p = 0 formulas, with arbitrary mixtures of AND and OR clauses); the only efficient algorithm for mixtures of AND, OR, and general integer-valued clauses; and tied for fastest for general Max 2-CSP (p = 1). Since a pure 2-Sat input instance may be transformed to a general CSP instance in the course of being solved, the algorithm's efficiency and generality go hand in hand. Our novel analysis results in a family of running-time bounds, each optimized for a particular value of p. The algorithm uses new reductions introduced here, as well as recent reductions such as “clause-learning” and “2-reductions” adapted to our setting's mixture of simple and general clauses. Each reduction imposes constraints on various parameters, and the running-time bound is an “objective function” of these parameters and p. The optimal running-time bound is obtained by solving a convex nonlinear program, which can be done efficiently and with a certificate of optimality.

We introduce a problem class we call Polynomial Constraint Satisfaction Problems, or PCSP. Where the usual CSPs from computer science and optimization have real-valued score functions, and partition functions from physics have monomials, PCSP has scores that are arbitrary multivariate formal polynomials, or indeed take values in an arbitrary ring. Although PCSP is much more general than CSP, remarkably, all (exact, exponential-time) algorithms we know of for 2-CSP (where each score depends on at most 2 variables) extend to 2-PCSP, at the expense of just a polynomial factor in running time. Specifically, we extend the reduction-based algorithm of Scott and Sorkin [2007]; the specialization of that approach to sparse random instances, where the algorithm runs in polynomial expected time; dynamic-programming algorithms based on tree decompositions; and the split-and-list matrix-multiplication algorithm of Williams [2004]. This gives the first polynomial-space exact algorithm more efficient than exhaustive enumeration for the well-studied problems of finding a maximum bisection of a graph, and calculating the partition function of an Ising model. It also yields the most efficient algorithm known for certain instances of counting and/or weighted Maximum Independent Set. Furthermore, PCSP solves both optimization and counting versions of a wide range of problems, including all CSPs, and thus enables samplers including uniform sampling of optimal solutions and Gibbs sampling of all solutions.

We consider the gap between the cost of an optimal assignment in a complete bipartite graph with random edge weights, and the cost of an optimal traveling salesman tour in a complete directed graph with the same edge weights. Using an improved “patching” heuristic, we show that with high probability the gap is $O((\ln n)^2/n)$, and that its expectation is $\Omega(1/n)$. One of the underpinnings of this result is that the largest edge weight in an optimal assignment has expectation $\Theta(\ln n / n)$. A consequence of the small assignment–TSP gap is an $e^{\tilde{O}(\sqrt{n})}$‐time algorithm which, with high probability, exactly solves a random asymmetric traveling salesman instance. In addition to the assignment–TSP gap, we also consider the expected gap between the optimal and second‐best assignments; it is at least $\Omega(1/n^2)$ and at most $O(\ln n/n^2)$.

Let F/sub 3/(n,m) be a random 3-SAT formula formed by selecting uniformly, independently and with replacement, m clauses among all 8(/sup n/C/sub 3/) possible 3-clauses over n variables. It has been conjectured that there exists a constant r/sub 3/ such that, for any /spl epsiv/>0, F/sub 3/[n,(r/sub 3/-/spl epsiv/)n] is almost surely satisfiable, but F/sub 3/[n,(r/sub 3/+/spl epsiv/)n] is almost surely unsatisfiable. The best lower bounds for the potential value of r/sub 3/ have come form analyzing rather simple extensions of unit-clause propagation. It was shown by D. Achlioptas (2000) that all these extensions can be cast in a common framework and analyzed in a uniform manner by employing differential equations. We determine optimal algorithms that are expressible in that framework, establishing r/sub 3/>3.26. We extend the analysis via differential equations, and make extensive use of a new optimization problem that we call the "max-density multiple-choice knapsack" problem. The structure of optimal knapsack solutions elegantly characterizes the choices made by an optimal algorithm.

We present a linear programming-based method for finding "gadgets," i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich, and Sudan [SIAM J. Comput., 27 (1998), pp. 804--915] of how to prove the optimality of gadgets: linear programming duality gives such proofs. The new gadgets, when combined with recent results of Håstad [ Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1--10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of $16/17 + ε$ and $12/13+ ε,$ respectively, is NP-hard for every ε > 0. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [ SIAM J. Comput., 27 (1998), pp. 804--915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and Håstad is 18/19. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801. This improves upon the previous best bound (implicit from M. X. Goemans and D. P. Williamson [J. ACM, 42 (1995), pp. 1115--1145]; U. Feige and M. X. Goemans [Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182--189]) of .7704.

The authors present a linear-programming based method for finding "gadgets", i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method they present a number of new, computer-constructed gadgets for several different reductions. This method also answers the question of how to prove the optimality of gadgets-they show how LP duality gives such proofs. The new gadgets improve hardness results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 60/61 and 44/45 respectively is NP-hard (improving upon the previous hardness of 71/72 for both problems). They also use the gadgets to obtain an improved approximation algorithm for MAX 3SAT which guarantees an approximation ratio of 0.801, This improves upon the previous best bound of 0.7704.

We resolve in the affirmative a question of R.B. Boppana and T. Bui: whether simulated annealing can with high probability and in polynomial time, find the optimal bisection of a random graph an G/sub npr/ when p-r=(/spl Theta/n/sup /spl Delta/-2/) for /spl Delta//spl les/2. (The random graph model G/sub npr/ specifies a "planted" bisection of density r, separating two n/2-vertex subsets of slightly higher density p.) We show that simulated "annealing" at an appropriate fixed temperature (i.e., the Metropolis algorithm) finds the unique smallest bisection in O(n/sup 2+/spl epsi//) steps with very high probability, provided /spl Delta/>11/6. (By using a slightly modified neighborhood structure, the number of steps can be reduced to O(n/sup 1+/spl epsi//).) We leave open the question of whether annealing is effective for /spl Delta/ in the range 3/2>

A two-dimensional stochastic model of the global wiring of a VLSI chip in a standard-cell or sea-of-gates design style is defined; prominent in the model is the property that the probability of connecting two pins is solely a function of the distance between the cells containing them. It is also assumed that each net consists of just two pins. A lower bound is placed on the expected size of the chip with the best possible wiring. An upper bound is placed on the expected size of the chip with a trivial (all randomly-oriented "L"s) wiring scheme. If the chip size is m rows by n columns and the size of the average row is /overbar μ/ the sizes of the trivial and perfect routings, expressed as a fraction of the size of the perfect routing, approaches 0 as √2 log (n)/ /overbar μ/ It is also shown that with probability at least 1 - ∊ size increase is no more than √2 log (mn/∊/ /overbar μ/.

The Planar Package Planner is a design aid aimed at helping to form a package layout plan, given only the information available during project initiation to digital system logic and package designers. A hierarchical approach is adopted, and a clustering program makes possible use of the layout scheme for bottom-up as well as top-down design. The layout plan for an experimental microprocessor is worked out as an example of the method.