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Dimitris Achlioptas

Publications and source records attributed to Dimitris Achlioptas.

29 records · Page 2Linked to original sources
arXiv2008

Algorithmic barriers from phase transitions

For many random Constraint Satisfaction Problems, by now, we have asymptotically tight estimates of the largest constraint density for which they have solutions. At the same time, all known polynomial-time algorithms for many of these problems already completely fail to find solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms already achieve this goal. Given the simplicity of those algorithms, one would expect there is a lot of room for improvement. Yet, to date, no algorithm is known that uses $(2-ε) χ$ colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we believe it is natural to inquire into its origin. We do so by analyzing the evolution of the set of $k$-colorings of a random graph, viewed as a subset of $\{1,...,k\}^{n}$, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly, the set of $k$-colorings looks like a giant ball for $k \ge 2 χ$, but like an error-correcting code for $k \le (2-ε) χ$. We prove that a completely analogous phase transition also occurs both in random $k$-SAT and in random hypergraph 2-coloring. And that for each problem, its location corresponds precisely with the point were all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to prove rigorously much of the celebrated 1-step Replica-Symmetry-Breaking hypothesis of statistical physics for random CSPs.

math.CO↗
arXiv2006

On the Solution-Space Geometry of Random Constraint Satisfaction Problems

For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomial-time algorithms for these problems fail to find solutions even at much lower densities. To understand the origin of this gap we study how the structure of the space of solutions evolves in such problems as constraints are added. In particular, we prove that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters. Moreover, inside each cluster most variables are frozen, i.e., take only one value. The existence of such frozen variables gives a satisfying intuitive explanation for the failure of the polynomial-time algorithms analyzed so far. At the same time, our results establish rigorously one of the two main hypotheses underlying Survey Propagation, a heuristic introduced by physicists in recent years that appears to perform extraordinarily well on random constraint satisfaction problems.

cs.CC↗
arXiv2006

On the Bias of Traceroute Sampling; or, Power-law Degree Distributions in Regular Graphs

Understanding the structure of the Internet graph is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining its graph structure is a surprisingly difficult task, as edges cannot be explicitly queried. Instead, empirical studies rely on traceroutes to build what are essentially single-source, all-destinations, shortest-path trees. These trees only sample a fraction of the network's edges, and a recent paper by Lakhina et al. found empirically that the resuting sample is intrinsically biased. For instance, the observed degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson. In this paper, we study the bias of traceroute sampling systematically, and, for a very general class of underlying degree distributions, calculate the likely observed distributions explicitly. To do this, we use a continuous-time realization of the process of exposing the BFS tree of a random graph with a given degree distribution, calculate the expected degree distribution of the tree, and show that it is sharply concentrated. As example applications of our machinery, we show how traceroute sampling finds power-law degree distributions in both delta-regular and Poisson-distributed random graphs. Thus, our work puts the observations of Lakhina et al. on a rigorous footing, and extends them to nearly arbitrary degree distributions.

cond-mat.dis-nn↗
arXiv2005

Rapid Mixing for Lattice Colorings with Fewer Colors

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.

cond-mat.stat-mech↗
arXiv2005

Hiding Satisfying Assignments: Two are Better than One

The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random k-SAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, as the formula's density increases, for a number of different algorithms, A acts as a stronger and stronger attractor. Motivated by recent results on the geometry of the space of satisfying truth assignments of random k-SAT and NAE-k-SAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and complement of A are satisfying. It appears that under this "symmetrization'' the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion.

cs.AI↗
arXiv2004

The Chromatic Number of Random Regular Graphs

Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the chromatic number is either k+1 or k+2.

cond-mat.dis-nn↗
arXiv2003

Random k-SAT: Two Moments Suffice to Cross a Sharp Threshold

Many NP-complete constraint satisfaction problems appear to undergo a "phase transition'' from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal k-SAT is 2^{k-1} ln 2 -O(1). As a corollary, we establish that the threshold for random k-SAT is of order Theta(2^k), resolving a long-standing open problem.

cond-mat.stat-mech↗
arXiv2003

The Threshold for Random k-SAT is 2^k ln2 - O(k)

Let F be a random k-SAT formula on n variables, formed by selecting uniformly and independently m = rn out of all possible k-clauses. It is well-known that if r>2^k ln 2, then the formula F is unsatisfiable with probability that tends to 1 as n tends to infinity. We prove that there exists a sequence t_k = O(k) such that if r < 2^k ln 2 - t_k, then the formula F is satisfiable with probability that tends to 1 as n tends to infinity. Our technique yields an explicit lower bound for the random k-SAT threshold for every k. For k>3 this improves upon all previously known lower bounds. For example, when k=10 our lower bound is 704.94 while the upper bound is 708.94.

cs.CC↗
arXiv2003

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 $δ_k \to 0$ such that if $r <(1-δ_k) R_k(p)$ then the probability that $F_k(n,rn)$is $p$-satisfiable tends to 1 as $n$ tends to infinity. The sequence $δ_k$ tends to 0 exponentially fast in $k$.

math.PR↗
arXiv2002

The Asymptotic Order of the k-SAT Threshold

Form a random k-SAT formula on n variables by selecting uniformly and independently m=rn clauses out of all 2^k (n choose k) possible k-clauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r_k such that, as n tends to infinity, the probability that the formula is satisfiable tends to 1 if r < r_k and to 0 if r > r_k. It has long been known that 2^k / k < r_k < 2^k. We prove that r_k > 2^{k-1} \ln 2 - d_k, where d_k \to (1+\ln 2)/2. Our proof also allows a blurry glimpse of the ``geometry'' of the set of satisfying truth assignments, and a nearly exact location of the threshold for Not-All-Equal (NAE) k-SAT.

cond-mat.stat-mech↗