Georgia Institute of Technology College of Sciences

Workshop Theme: Submodular functions are discrete analogues of convex functions, arising in various fields of computer science and operations research. Since the seminal work of Jack Edmonds (1970), submodularity has long been recognized as a common structure of many efficiently solvable combinatorial optimization problems. Recent algorithmic developments in the past decade include combinatorial strongly polynomial algorithm for minimization, constant factor approximation algorithms for maximization, and efficient methods for learning submodular functions. In addition, submodular functions find novel applications in combinatorial auctions, machine learning, and social networks. This workshop aims at providing a forum for researchers from a variety of backgrounds for exchanging results, ideas, and problems on submodular optimization and its applications. The workshop will be held from March 19-22, 2012. For complete details and workshop program, please see the website.

I present a new class of vertex cover and set cover games, with the price of anarchy bounds matching the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs. In particular, the price of anarchy is 2 for vertex cover. The basic intuition is that the members of the vertex cover form a Mafia that has to "protect" the graph, and may ask ransoms from their neighbors in exchange for the protection. These ransoms turn out to capture a good dual solution to the linear programming relaxation. For linear costs we also exhibit linear time best response dynamics that converge that mimic the classical greedy approximation algorithm of Bar-Yehuda and Even. This is a joint work with Georgios Piliouras and Tomas Valla.

Erd\H{o}s and Rothschild asked to estimate the maximum number, denotedby $h(n,c)$, such that every $n$-vertex graph with at least $cn^2$edges, each of which is contained in at least one triangle, mustcontain an edge that is in at least $h(n,c)$ triangles. In particular,Erd\H{o}s asked in 1987 to determine whether for every $c>0$ there is$\epsilon>0$ such that $h(n,c)>n^{\epsilon}$ for all sufficientlylarge $n$. We prove that $h(n,c)=n^{O(1/\log \log n)}$ for every fixed$c<1/4$. This gives a negative answer to the question of Erd\H{o}s,and is best possible in terms of the range for $c$, as it is knownthat every $n$-vertex graph with more than $n^2/4$ edges contains anedge that is in at least $n/6$ triangles.Joint work with Jacob Fox.

The fields of statistical physics, discrete probability, combinatorics, and theoretical computer science have converged around efforts to understand random structures and algorithms. Recent activity in the interface of these fields has enabled tremendous breakthroughs in each domain and has supplied a new set of techniques for researchers approaching related problems. This thesis makes progress on several problems in this interface whose solutions all build on insights from multiple disciplinary perspectives. First, we consider a dynamic growth process arising in the context of DNA-based self-assembly. The assembly process can be modeled as a simple Markov chain. We prove that the chain is rapidly mixing for large enough bias in regions of Z^d. The proof uses a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile, which arises in the nanotechnology application. Moreover, we use intuition from statistical physics to construct a choice of the biases for which the Markov chain M_{mon} requires exponential time to converge. Second, we consider a related problem regarding the convergence rate of biased permutations that arises in the context of self-organizing lists. The Markov chain M_{nn} in this case is a nearest-neighbor chain that allows adjacent transpositions, and the rate of these exchanges is governed by various input parameters. It was conjectured that the chain is always rapidly mixing when the inversion probabilities are positively biased, i.e., we put nearest neighbor pair x < y in order with bias 1/2 <= p_{xy} <= 1 and out of order with bias 1-p_{xy}. The Markov chain M_{mon} was known to have connections to a simplified version of this biased card-shuffling. We provide new connections between M_{nn} and M_{mon} by using simple combinatorial bijections, and we prove that M_{nn} is always rapidly mixing for two general classes of positively biased {p_{xy}}. More significantly, we also prove that the general conjecture is false by exhibiting values for the p_{xy}, with 1/2 <= p_{xy} <= 1 for all x < y, but for which the transposition chain will require exponential time to converge. Finally, we consider a model of colloids, which are binary mixtures of molecules with one type of molecule suspended in another. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters. This clustering has proved elusive to verify, since all local sampling algorithms are known to be inefficient at high density, and in fact a new nonlocal algorithm was recently shown to require exponential time in some cases. We characterize the high and low density phases for a general family of discrete interfering binary mixtures by showing that they exhibit a "clustering property" at high density and not at low density. The clustering property states that there will be a region that has very high area, very small perimeter, and high density of one type of molecule. Special cases of interfering binary mixtures include the Ising model at fixed magnetization and independent sets.