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

A simple and sharper proof of the hypergraph Moore bound

The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of hyperedges in a hypergraph. For graphs (i.e., $2$-uniform hypergraphs), a bound tight up to the leading constant was proven in a classical work of Alon, Hoory and Linial [AHL02]. For hypergraphs of uniformity $k>2$, an appropriate generalization was conjectured by Feige [Fei08]. The conjecture was settled up to an additional $\log^{4k+1} n$ factor in the size in a recent work of Guruswami, Kothari and Manohar [GKM21]. Their argument relies on a connection between the existence of short even covers and the spectrum of a certain randomly signed Kikuchi matrix. Their analysis, especially for the case of odd $k$, is significantly complicated. In this work, we present a substantially simpler and shorter proof of the hypergraph Moore bound. Our key idea is the use of a new reweighted Kikuchi matrix and an edge deletion step that allows us to drop several involved steps in [GKM21]'s analysis such as combinatorial bucketing of rows of the Kikuchi matrix and the use of the Schudy-Sviridenko polynomial concentration. Our simpler proof also obtains tighter parameters: in particular, the argument gives a new proof of the classical Moore bound of [AHL02] with no loss (the proof in [GKM21] loses a $\log^3 n$ factor), and loses only a single logarithmic factor for all $k>2$-uniform hypergraphs. As in [GKM21], our ideas naturally extend to yield a simpler proof of the full trade-off for strongly refuting smoothed instances of constraint satisfaction problems with similarly improved parameters

System Verification and Runtime Monitoring with Multiple Weakly-Hard Constraints

A weakly-hard fault model can be captured by an (m,k) constraint, where 0≤ m ≤ k , meaning that there are at most m bad events (faults) among any k consecutive events. In this article, we use a weakly-hard fault model to constrain the occurrences of faults in system inputs. We develop approaches to verify properties for all possible values of (m,k) , where k is smaller than or equal to a given  K , in an exact and efficient manner. By verifying all possible values of (m,k) , we define weakly-hard requirements for the system environment and design a runtime monitor based on counting the number of faults in system inputs. If the system environment satisfies the weakly-hard requirements, then the satisfaction of desired properties is guaranteed; otherwise, the runtime monitor can notify the system to switch to a safe mode. This is especially essential for cyber-physical systems that need to provide guarantees with limited resources and the existence of faults. Experimental results with discrete second-order control, network routing, vehicle following, and lane changing demonstrate the generality and the efficiency of the proposed approaches. </jats:p

Measurement of the WW Boson Mass

A measurement of the mass of the $W$ boson is presented based on a sample of 5982 $W \rightarrow e \nu$ decays observed in $p\overline{p}$ collisions at $\sqrt{s}$ = 1.8~TeV with the D\O\ detector during the 1992--1993 run. From a fit to the transverse mass spectrum, combined with measurements of the $Z$ boson mass, the $W$ boson mass is measured to be $M_W = 80.350 \pm 0.140 (stat.) \pm 0.165 (syst.) \pm 0.160 (scale) GeV/c^2$.Comment: 12 pages, LaTex, style Revtex, including 3 postscript figures (submitted to PRL