On the Maximum Satisfiability of Random Formulas

Maximum satisfiability is a canonical NP-hard optimization problem that appears empirically hard for random instances. Let us say that a Conjunctive normal form (CNF) formula consisting of $k$-clauses is $p$-satisfiable if there exists a truth assignment satisfying $1-2^{-k}+p 2^{-k}$ of all clauses (observe that every $k$-CNF is 0-satisfiable). Also, let $F_k(n,m)$ denote a random $k$-CNF on $n$ variables formed by selecting uniformly and independently $m$ out of all possible $k$-clauses. It is easy to prove that for every $k>1$ and every $p$ in $(0,1]$, there is $R_k(p)$ such that if $r >R_k(p)$, then the probability that $F_k(n,rn)$ is $p$-satisfiable tends to 0 as $n$ tends to infinity. We prove that there exists a sequence $\delta_k \to 0$ such that if $r <(1-\delta_k) R_k(p)$ then the probability that $F_k(n,rn)$is $p$-satisfiable tends to 1 as $n$ tends to infinity. The sequence $\delta_k$ tends to 0 exponentially fast in $k$

Counting in two-spin models on d-regular graphs

We establish that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting “free energy density” which coincides with the (nonrigorous) Bethe prediction of statistical physics. Using this result, we characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs without the use of the second moment method employed in previous works on these questions. As a consequence, we show that for both the hard-core model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on d-regular graphs when the model has nonuniqueness on the d-regular tree. Together with results of Jerrum–Sinclair, Weitz, and Sinclair–Srivastava– Thurley, this gives an almost complete classification of the computational complexity of homogeneous two-spin systems on bounded-degree graphs.Supported in part by Alfred P. Sloan Research Fellowship. Supported in part by Department of Defense NDSEG Fellowships

Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end

We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses ($p$-spin model), and $k$-local constraint satisfaction problems ($k$-CSP). We show that either (a) the algorithm finds the optimal solution in time $O^*(2^{(0.5-c)n})$ for an $n$-independent constant $c$, a $2^{cn}$ advantage over Grover's algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a $(1-\eta)$ approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of $\eta$. Additionally, we show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case. The algorithm and its analysis is largely inspired by Hastings' short-path algorithm [$\textit{Quantum}$ $\textbf{2}$ (2018) 78].Comment: 49 pages, 3 figure

Algorithms and algorithmic obstacles for probabilistic combinatorial structures

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 209-214).We study efficient average-case (approximation) algorithms for combinatorial optimization problems, as well as explore the algorithmic obstacles for a variety of discrete optimization problems arising in the theory of random graphs, statistics and machine learning. In particular, we consider the average-case optimization for three NP-hard combinatorial optimization problems: Large Submatrix Selection, Maximum Cut (Max-Cut) of a graph and Matrix Completion. The Large Submatrix Selection problem is to find a k x k submatrix of an n x n matrix with i.i.d. standard Gaussian entries, which has the largest average entry. It was shown in [13] using non-constructive methods that the largest average value of a k x k submatrix is 2(1 + o(1) [square root] log n/k with high probability (w.h.p.) when k = O(log n/ log log n). We show that a natural greedy algorithm called Largest Average Submatrix LAS produces a submatrix with average value (1+ o(1)) [square root] 2 log n/k w.h.p. when k is constant and n grows, namely approximately [square root] 2 smaller. Then by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a k x k matrix with asymptotically the same average value (1+o(1) [square root] 2log n/k w.h.p., for k = o(log n). Since the maximum clique problem is a special case of the largest submatrix problem and the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor [square root] 2 performance gap suffered by both algorithms might be very challenging. Surprisingly, we show the existence of a very simple algorithm which produces a k x k matrix with average value (1 + o[subscript]k(1) + o(1))(4/3) [square root] 2log n/k for k = o((log n)¹.⁵), that is, with asymptotic factor 4/3 when k grows. To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically (1 + o(1))[alpha][square root] 2 log n/k for a fixed value [alpha] [epsilon] [1, fixed value a E [1, [square root]2]. The overlap corresponds to the number of common rows and common columns for pairs of matrices achieving this value. We discover numerically an intriguing phase transition at [alpha]* [delta]= 5[square root]2/(3[square root]3) ~~ 1.3608.. [epsilon] [4/3, [square root]2]: when [alpha] [alpha]*, appropriately defined. We conjecture that OGP observed for [alpha] > [alpha]* also marks the onset of the algorithmic hardness - no polynomial time algorithm exists for finding matrices with average value at least (1+o(1)[alpha][square root]2log n/k, when [alpha] > [alpha]* and k is a growing function of n. Finding a maximum cut of a graph is a well-known canonical NP-hard problem. We consider the problem of estimating the size of a maximum cut in a random Erdős-Rényi graph on n nodes and [cn] edges. We establish that the size of the maximum cut normalized by the number of nodes belongs to the interval [c/2 + 0.47523[square root]c,c/2 + 0.55909[square root]c] w.h.p. as n increases, for all sufficiently large c. We observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved multi-dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as c increases and use it to obtain an improved upper bound on the Max-Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value c/2 + 0.47523[square root]c. We also obtain an improved lower bound of 1.36000n on the Max-Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of 1.33773n. Matrix Completion is the problem of reconstructing a rank-k n x n matrix M from a sampling of its entries. We propose a new matrix completion algorithm using a novel sampling scheme based on a union of independent sparse random regular bipartite graphs. We show that under a certain incoherence assumption on M and for the case when both the rank and the condition number of M are bounded, w.h.p. our algorithm recovers an [epsilon]-approximation of M in terms of the Frobenius norm using O(nlog² (1/[epsilon])) samples and in linear time O(nlog² (1/[epsilon])). This provides the best known bounds both on the sample complexity and computational cost for reconstructing (approximately) an unknown low-rank matrix. The novelty of our algorithm is two new steps of thresholding singular values and rescaling singular vectors in the application of the "vanilla" alternating minimization algorithm. The structure of sparse random regular graphs is used heavily for controlling the impact of these regularization steps.by Quan Li.Ph. D