### DIMACS Workshop on Phase Transitions in Random Structures and Algorithms

#### March 19 - 23, 2007 Klaus Advanced Computing Building Klaus Main Auditorium, room 1443 Georgia Institute of Technology

Organizers:
Gregory Sorkin, IBM Research, sorkin@watson.ibm.com
Eric Vigoda, Georgia Institute of Technology, vigoda@cc.gatech.edu
Presented under the auspices of the Special Focus on Discrete Random Systems.

#### Abstracts:

Dimitris Achlioptas, UC Santa Cruz

Title: On the solution space geometry of random CSPs

For a number of random constraint satisfaction problems, such as random k-SAT and random graph coloring, we now have excellent estimates of the largest constraints-to-variables ratio for which they have solutions. These estimates imply that all known polynomial- time algorithms stop finding solutions much before solutions cease to exist. To understand the origin of this phenomenon we study the evolution of the structure of the space of solutions in random CSPs as constraints are added. In particular, we will survey both what physicists hypothesize happens in this evolution, and what has been rigorously proven so far. We will see that the rigorous results are both in agreement with the physics predictions and help demystify Survey Propagation, an extremely successful heuristic proposed by physicists for random CSPs.

Marek Biskup, UCLA

Title: Anomalous heat kernel decay for random walk among random conductances

I will consider the random walk on Z^d driven by a field of random i.i.d. conductances. The law of the conductances is bounded from above; no restriction is posed on the lower tail (at zero) except that the bonds with positive conductances percolate. The presence of very weak bonds allows the random walk in a finite box mix pretty much arbitrarily slowly. However, when we focus attention on the return probability to the starting point -- i.e., the heat-kernel -- it turns out that in dimensions d=2,3 the decay is as for the simple random walk. On the other hand, in d>4 the heat- kernel at time 2n may decay as slowly as o(1/n^2) and in d=4 as slowly as O(n^{-2}log n). These upper bounds can be matched arbitrarily closely by lower bounds in particular examples. Despite this, the random walk scales to Brownian motion under the usual diffusive scaling of space and time.

Based on joint works with N. Berger, C. Hoffman, G. Kozma and T. Prescott.

Béla Bollobás, Cambridge University and University of Memphis

Title: Percolation on sequences of dense graphs

Christian Borgs, Microsoft Research

Title: Sharp Threshold for Percolation on Convergent Graph Sequences

Jennifer Chayes, Microsoft Research

Title: First to Market is not Everything: Phase Transitions in Preferential Attachment with Fitness

Amin Coja-Oghlan, Humboldt University

Title: Local limit theorems for the giant component

Abraham Flaxman, Microsoft Research

Title: Expansion and lack thereof in randomly perturbed graphs

This talk will investigate the expansion properties of randomly perturbed graphs. These graphs are formed by, for example, adding a random 1-out or very sparse Erdos-Renyi graph to an arbitrary connected graph. When any connected n-vertex base graph is perturbed by adding a random 1-out then, with high probability, the resulting graph has expansion properties. When the perturbation is by a sparse Erdos-Renyi graph, the expansion of the perturbed graph depends on the structure of the base graph. The same techniques also apply to bound the expansion in the small worlds graphs described by Watts and Strogatz in [Nature 292 (1998), 440--442] and by Kleinberg in [Proc. of 32nd ACM Symposium on Theory of Computing (2000), 163--170]. Analysis of Kleinberg's model reveals a phase transition: the graph stops being an expander exactly at the point where a decentralized algorithm is effective in constructing a short path.

The proofs of expansion rely on a way of summing over subsets of vertices which allows an argument based on the First Moment Method to succeed.

Alan Frieze, CMU

Title: The cover time of random graphs

We give asymptotically precise estimates for the cover time of random graphs and digraphs drawn from various distributions: Random Regular Graphs; G(n,p); The Giant Component of G(n,p); The Preferential Attachment Graph; The Random Digraph D(n,p).

Joint work with Colin Cooper

David Gamarnik, MIT

Title: Monomer-dimer model and a new deterministic approximation algorithm for computing a permanent of a 0,1 matrix.

We construct a deterministic approximation algorithm for computing a permanent of a 0,1 matrix to within a multiplicative factor $(1+eps)^n$, for arbitrary positive constant eps. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general case the running time of the algorithm is sub exponential $e^{O( n^{2/3}\log^3 n)}$ For the class of graphs which are constant degree expanders the first result is an improvement over the best known approximation factor $e^n$.

Our results use a recent deterministic approximation algorithm for computing a partition function of a monomer-dimer model, and Jerrum-Vazirani expander decomposition method. Joint work with Dmitriy Katz

Leslie Ann Goldberg, University of Liverpool

Title: The Complexity of Weighted Boolean #CSP

This work gives a dichotomy theorem for the complexity of exactly computing the partition function of a weighted Boolean constraint satisfaction instance. The problem is parameterised by a fixed "constraint language" which is the set of weighted relations that may be used to constrain an instance. We show that the partition function can be computed in polynomial time if either
(1)every weighted relation in the constraint language is "of product type", or
( 2)every weighted relation in the constraint language is "pure affine". For every other fixed constraint language, computing the partition function is #P-hard.

Joint work with Martin Dyer and Mark Jerrum.

Mark Jerrum, Queen Mary

Title: Inapproximability of the Tutte polynomial

The (classical) Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes much information about G. The number of spanning trees in G, the number of acyclic orientations of G and the partition function of the q-state Potts model are all specialisations of the Tutte polynomial. From a complexity-theoretic point of view, "mapping the Tutte plane" amounts to determining the computational complexity of evaluating T(G;x,y), given G, for each rational pair (x,y). For exact computation, the mapping was done in detail by Jaeger, Vertigan and Welsh. For approximate computation (within specified relative errormuch less was known. I'll report on recent work with Leslie Goldberg (Liverpool) that makes some progress in this direction.

Andrea Montanari, Stanford

Title : Belief Propagation, Cavity Method and Pure Gibbs States in Combinatorial Problems

Consider the uniform measure over satisfying assignments of a k-satisfiability formula, or the size-weigthed measure over independent sets of a graph. Belief propagation and the cavity method provide heuristic estimates for the marginals of such distribution. They both are based on a remarkable assumption on the structure of the corresponding measure, prescribing a precise form of (approximate) factorization according to the underlying graph.

I will discuss conditions for the assumption to hold. This will bring up the connection with pure states, and `clusters' of solutions in constraint satisfaction problems.

Thierry Mora, Université Paris-Sud

Title: Clustering and pairs of solutions in random Boolean problems

One of the most striking predictions of the statistical physics analysis of random constraint satisfactions problems is the existence of a clustering phenomenon, by which the space of solutions divides itself into many connected components far from each other. In order to investigate this geometrical property, we introduce the notion of x-satisfiability: a problem is x-satisfiable if there exist two solutions separated by a relative distance x, i.e. differing by a proportion x of variables. Using first and second moment methods, I will show how the existence of a gap in the distance spectrum can be proved rigorously in the k-satisfiability problem, for k large enough. This means that one can find pairs of solutions close to each other, and around x=1/2, but that a whole range of intermediate distances is inaccessible. I will then show how to yield an arguably exact value of the x-satisfiability threshold in the simpler case of random k-XORSAT using message-passing techniques.

Boris Pittel, Ohio State University

Title: On a random graph evolving by degrees

We consider a random (multi)graph process $\{G_m\}$ on a vertex set $[n]$, which is the special case of a more general process put forth by Laci Lov\'asz few years ago. $G_0$ is empty, and $G_{m+1}$ is obtained from $G_m$ by inserting a new edge $e$ at random. Specifically, the conditional probability that $e$ joins two currently disjoint vertices, $i$ and $j$, is proportional to $, d_i+\alpha), d_j+\alpha)$, where $d_i$, $d_j$ are the degrees of $i$, $j$ in $G_m$, and $\alpha>0$ is a fixed parameter. The limiting case $\alpha=\infty$ is the Erd\H os-R\'enyi graph process. We show that whp $G_m$ contains a unique giant component iff $c:=2m/n > c_{\alpha}= \alpha/, 1+\alpha)$, and the size of this giant is asymptotic to $$n\left[1-\left, \frac{\alpha+c^*}{\alpha+c}\right)^{\alpha}\right],$$ where $c^*1$, $m_n$ is the threshold for connectedness of $G_m$ itself.

Luis Rademacher, MIT

Title: Dispersion of Mass and Lower Bounds for Randomized Algorithms

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.

Daniel Stefankovic, University of Rochester

Title: Adaptive Annealing: A Near-optimal Connection between Sampling and Counting

We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. Our result is set in the framework of approximating the partition function $Z$ of a discrete system, such as the Ising model, the matchings, or colorings of a graph. The typical approach to estimating the partition function $Z(\beta)$ at some desired (inverse) temperature $\beta^*$ is to define a sequence, which we call a {\em cooling schedule}, $\beta_0=0<\beta_1<\dots<\beta_\ell=\beta^*$ where $Z(0)$ is trivial to compute and the ratios $Z (\beta_{i+1})/Z(\beta_i)$ are easy to estimate by sampling from the distribution corresponding to $Z(\beta_i)$. Previous approaches required a cooling schedule of length $O^*(\ln{A})$ where $A=Z(0)$, thereby ensuring that each ratio $Z(\beta_{i+1})/Z(\beta_i)$ is bounded. We present a cooling schedule of length $O^*(\sqrt{\ln{A}})$. For well-studied problems such as estimating the partition function of the Ising model, or approximating the number of $k$-colorings or matchings of a graph, our cooling schedule is of length $O^*( \sqrt{n})$, which yields an overall savings of $O^*( n)$ in the running time of the approximate counting algorithm compared to previous results.

Joint work with Santosh Vempala and Eric Vigoda.

Jeff Steif, Chalmers

Title: Stochastic domination and the Ising model

(1). We show that the plus and minus states for the Ising model on Z^d dominate the same set of product measures. We show that this latter fact however completely fails on the homogenous 3-ary tree. (2). While it is known that the plus states for different temperatures on Z^d are never stochastically ordered, on the homogenous 3-ary tree, almost the complete opposite is the case. (3). On Z^d, the set of product measures which the plus state for the Ising model dominates is strictly decreasing in the interaction parameter. (4). An FKG exchangeable finite process dominates a product measure if and only the relevant inequality holds for the event of all 0's. (5). Extending these results to an amenable/nonamenable dichotomy would be interesting. This is based on joint work with Tom Liggett.

Yuval Peres, Microsoft Research and UC Berkeley

Title: Critical random graphs: diameter and mixing time

Let C_1 denote the largest connected component of the critical Erdos-Renyi random graph G(n,1/n). We show that, typically, the diameter of C_1 is of order n^{1/3} and the mixing time of the lazy simple random walk on C_1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size n^{2/3} of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d-1\leq 1.

Joint work with Asaf Nachmias.

Sebastien Roch, UC Berkeley

Title: Markov Models on Trees: Reconstruction and Applications

Markov models on trees arise naturally in many fields, notably in molecular biology - as models of evolution; in statistical physics - as models of spin systems; and in networking - as models of broadcasting. In this talk, I will discuss various inference problems motivated especially by applications in statistical phylogenetics, i.e. the reconstruction of evolutionary histories of organisms from their molecular sequences. In particular, I will consider the "root reconstruction" problem: how accurately can one guess the value at the root of the tree, given the state at the leaves? I will focus on recent work establishing new conditions for the impossibility of such reconstruction. I will also discuss the related "phylogenetic reconstruction" problem: given enough samples at the leaves, can one reconstruct the tree that generated this data and, if so, how efficiently? I will present a recent result on a sharp transition in the number of samples required to recover the tree topology, using a connection to the root reconstruction problem above. Time permitting, I will describe briefly connections to computational learning theory and network tomography as well.

This is joint work with S. Bhamidi, C. Borgs, J. Chayes, C. Daskalakis, E. Mossel, and R. Rajagopal.

Sekhar Tatikonda, Yale

Title: Mixing and clustering in message-passing schemes - A Gibbs measure view

The belief propagation algorithm and its variants have recently been shown to work quite well on hard constraint satisfaction problems. Survey propagation is one such algorithm that has been developed for solving the k-SAT problem. This algorithm is based on two hypotheses deriving from the replica symmetry breaking theory developed for the statistical mechanics of spin glasses. The first is that the solutions of constraint satisfaction problems near threshold organize into disjoint clusters. The second is that within each cluster there is spatial mixing. In this paper we show that both hypotheses hold for a large class of constraint satisfaction problems including the k-SAT problem. Our approach is based on examining an appropriate infinite site Gibbs measure. Each cluster corresponds to a different extremal measure. Thus the so-called greedy threshold is equivalent to the Gibbs measure uniqueness threshold. Below this threshold there is one cluster and above it there are many.

Prasad Tetali, Georgia Tech

Title: Random walks for the discrete log problem

We analyze the classical Pollard's Rho algorithm for finding the discrete logarithms in a cyclic group $G$. We prove that, with high probability, a collision occurs (and the discrete logarithm found) in O(\sqrt{|G|\log|G|\log\log|G|}) steps, by analyzing an appropriate (nonreversible, nonlazy) random walk on a discrete cycle of odd length.

This is joint work with Jeong Han Kim and Ravi Montenegro

Juan C. Vera, Georgia Tech

Title: Randomly coloring planar graphs with fewer colors than the maximum degree

We study Markov chains for randomly sampling k-colorings of a graph with maximum degree D. We prove the first results, for a general class of graphs, showing fast convergence of such chains when the number of colors k << D. When D =O((log n)^(1+eps))and k > D/log D we prove O(n log n) mixing time of the single-site update chain, known as the Glauber dynamics, for any planar graph. A challenging aspect of the case when D is constant and k < D is that, even in a random coloring, a constant fraction of the vertices are frozen, with a unique color not present among its neighbors. To extend our results to this setting, we introduce a new Markov chain which recolors vertices in a partially deterministic order defined in terms of the principal eigenvector of the graph. For this chain we prove nearly-linear time convergence when k > D/ log D. This implies polynomial mixing time of the original Glauber dynamics under the same conditions.

Joint work with Thomas P. Hayes and Eric Vigoda

Santosh Vempala, Georgia Tech

Title: The Complexity of Volume Computation: Annealing vs Dispersion

Computing the volume of a convex body in n-dimensional space is a basic problem that has lead to many nice techniques. In this talk, we survey the algorithmic and geometric ideas behind the most recent developments in sampling, integration and optimization over convex bodies and more generally, logconcave functions. In particular, we will discuss the analysis of the random walk called "hit-and-run", and simulated annealing in the continuous setting which together lead to the current best algorithms for both integration and optimization. We'll briefly touch upon possible future improvements given recent dispersion-based lower bounds.

Parts of the talk are joint work with Laci Lovasz and with Luis Rademacher.

Van Vu, Rutgers

Title: Monotonicity and Resilience of random graphs

(1) Does it make sense to talk about thresholds for random regular graphs? In other words, is the random regular graphs model monotone? (If one tells us that a random n^{1/2}-regular graph has a.s a monotone property P, can we conclude that a random n/2-regular graph has a.s. the same property?

(2) The resilience of a graph towards a specific property. Assume that G has a property P, how much should we change G in order to destroy P? Different notions of resilience: Global and Local resilience. The resilience of a random graphs with respect to the most basic properties.

(3) The relation between (1) and ( 2).

(4)The resilience of other random structures. Open questions.

Based on joined work with J. H. Kim and B. Sudakov.

Peter Winkler, Dartmouth

Title: Coordinate Percolation

In coordinate percolation, the life or death of a site is determined by events associated with the lines or planes through the point. We will review some examples and show some results (with Lizz Moseman) on a particularly tractable form.

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Document last modified on March 1, 2007.