Certifying solution geometry in random CSPs: counts, clusters and balance

An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following question: what does the solution space for a random CSP look like to an efficient algorithm? In pursuit of this inquiry, we focus on the following problems about random Boolean CSPs at the densities where they are unsatisfiable but no refutation algorithm is known. 1. Counts. For every Boolean CSP we give algorithms that with high probability certify a subexponential upper bound on the number of solutions. We also give algorithms to certify a bound on the number of large cuts in a Gaussian-weighted graph, and the number of large independent sets in a random $d$-regular graph. 2. Clusters. For Boolean $3$CSPs we give algorithms that with high probability certify an upper bound on the number of clusters of solutions. 3. Balance. We also give algorithms that with high probability certify that there are no "unbalanced" solutions, i.e., solutions where the fraction of $+1$s deviates significantly from $50\%$. Finally, we also provide hardness evidence suggesting that our algorithms for counting are optimal

New Notions and Constructions of Sparsification for Graphs and Hypergraphs

A sparsifier of a graph G (Benczu´r and Karger; Spielman and Teng) is a sparse weighted subgraph ˜ G that approximately retains the same cut structure of G. For general graphs, non-trivial sparsification is possible only by using weighted graphs in which different edges have different weights. Even for graphs that admit unweighted sparsifiers (that is, sparsifiers in which all the edge weights are equal to the same scaling factor), there are no known polynomial time algorithms that find such unweighted sparsifiers. We study a weaker notion of sparsification suggested by Oveis Gharan, in which the number of cut edges in each cut (S, ¯ S) is not approximated within a multiplicative factor (1 + ǫ), but is, instead, approximated up to an additive term bounded by ǫ times d · |S| + vol(S), where d is the average

New notions and constructions of sparsification for graphs and hypergraphs

No abstract availabl

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

Metastability cascades and prewetting in the SOS model

We study Glauber dynamics for the low temperature $(2+1)$D Solid-On-Solid model on a box of side-length $n$ with a floor at height $0$ (inducing entropic repulsion) and a competing bulk external field $\lambda$ pointing down (the prewetting problem). In 1996, Cesi and Martinelli showed that if the inverse-temperature $\beta$ is large enough, then along a decreasing sequence of critical points $(\lambda_c^{(k)})_{k=0}^{K_\beta}$ the dynamics is torpid: its inverse spectral gap is $O(1)$ when $\lambda \in (\lambda_c^{(k+1)},\lambda_c^{(k)})$ whereas it is $\exp[\Theta(n)]$ at each $\lambda_c^{(k)}$ for each $k\leq K_\beta$, due to a coexistence of rigid phases at heights $k+1$ and $k$. Our focus is understanding (a) the onset of metastability as $\lambda_n\uparrow\lambda_c^{(k)}$; and (b) the effect of an unbounded number of layers, as we remove the restriction $k\le K_\beta$, and even allow for $\lambda_n\to 0$ towards the $\lambda = 0$ case which has $O(\log n)$ layers and was studied by Caputo et al. (2014). We show that for any $k$, possibly growing with $n$, the inverse gap is $\exp[\tilde\Theta(1/|\lambda_n-\lambda_c^{(k)}|)]$ as $\lambda\uparrow \lambda_c^{(k)}$ up to distance $n^{-1+o(1)}$ from this critical point, due to a metastable layer at height $k$ on the way to forming the desired layer at height $k+1$. By taking $\lambda_n = n^{-\alpha}$ (corresponding to $k_n\asymp \log n$), this also interpolates down to the behavior of the dynamics when $\lambda =0$. We compliment this by extending the fast mixing to all $\lambda$ uniformly bounded away from $(\lambda_c^{(k)})_{k=0}^\infty$. Together, these results provide a sharp understanding of the predicted infinite sequence of dynamical phase transitions governed by the layering phenomenon.Comment: 49 pages, 3 figure