David Gamarnik

Simple deterministic approximation algorithms for counting matchings

We construct a deterministic fully polynomial time approximationscheme (FPTAS) for computing the total number of matchings in abounded degree graph. Additionally, for an arbitrary graph, weconstruct a deterministic algorithm for computing approximately thenumber of matchings within running time exp(O(√n log2n)),where n is the number of vertices. Our approach is based on the correlation decay technique originating in statistical physics. Previously thisapproach was successfully used for approximately counting thenumber of independent sets and colorings in some classes of graphs [1, 24, 6].Thus we add another problem to the small, but growing, class of P-complete problems for whichthere is now a deterministic FPTAS.

Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

We establish the existence of free energy limits for several combinatorial models on Erdos–Renyi graph G(N,⌊cN⌋) and random r-regular graph G(N,r). For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239–298 Cambridge Univ. Press]; Bollobas and Riordan [Random Structures Algorithms 39 (2011) 1–38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259–264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1–72 Springer]. Our approach is based on extending and simplifying the interpolation method of Guerra and Toninelli [Comm. Math. Phys. 230 (2002) 71–79] and Franz and Leone [J. Stat. Phys. 111 (2003) 535–564]. Among other applications, this method was used to prove the existence of free energy limits for Viana–Bray and K-SAT models on Erdos–Renyi graphs. The case of zero temperature was treated by taking limits of positive temperature models. We provide instead a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of Erdos–Renyi graph G(N,⌊cN⌋) and random regular graph G(N,r). In addition we establish the large deviations principle for the satisfiability property of the constraint satisfaction problems, coloring, K-SAT and NAE-K-SAT, for the G(N,⌊cN⌋) random graph model.

Stability conditions for multiclass fluid queueing networks

We introduce a new method to investigate stability of work-conserving policies in multiclass queueing networks. The method decomposes feasible trajectories and uses linear programming to test stability. We show that this linear program is a necessary and sufficient condition for the stability of all work-conserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary. Previous research had identified sufficient conditions through the use of a particular class of (piecewise linear convex) Lyapunov functions. Using linear programming duality, we show that for two-station systems the Lyapunov function approach is equivalent to ours and therefore characterizes stability exactly.

Stability of adaptive and non-adaptive packet routing policies in adversarial queueing networks

We investigate the stability of packet routing policies in adversarial queueing networks. We provide a simple classification of networks which are stable under any greedy scheduling policy. We show that a network is stable if and only if the underlying undirected connected graph contains at most two edges. We also propose a simple and distributed policy which is stable in an arbitrary adversarial queueing network even for the critical value of the arrival rate r = 1. Finally, a simple and checkable network flow-type load condition is formulated for adaptive adversarial queueing networks, and a policy is proposed which achieves stability under this new load condition. This load condition is a relaxation of the integral network flow-type condition considered previously in the literature.

Counting without sampling: new algorithms for enumeration problems using statistical physics

We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by recent developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide ∈-approximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, unlike Markov chain based algorithms, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4-regular n-node graph with large girth has asymptotically (1.494 ...)n independent sets, and in every r-regular graph with n nodes and large girth the number of q ≥ r + 1-proper colorings is asymptotically (q(1-1/q)r/2)n for large n. In statistical physics terminology, we compute explicitly the partition function (free energy) in these cases. We extend our results to random regular graphs graphs also. The explicit results obtained in this paper would be hard to derive via Markov chain sampling technique.

An improved upper bound for the TSP in cubic 3-edge-connected graphs

We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs. These graphs are interesting because of the connection between their optimal solutions and the subtour elimination LP relaxation. Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP in general.

Finding long chains in kidney exchange using the traveling salesman problem

Significance There are currently more than 100,000 patients on the waiting list in the United States for a kidney transplant from a deceased donor. To address this shortage, kidney exchange programs allow patients with living incompatible donors to exchange donors through cycles and chains initiated by altruistic nondirected donors. To determine which exchanges will take place, kidney exchange programs use algorithms for maximizing the number of transplants under constraints about the size of feasible exchanges. This problem is NP-hard, and algorithms previously used were unable to optimize when chains could be long. We developed two algorithms that use integer programming to solve this problem, one of which is inspired by the traveling salesman, that together can find optimal solutions in practice. As of May 2014 there were more than 100,000 patients on the waiting list for a kidney transplant from a deceased donor. Although the preferred treatment is a kidney transplant, every year there are fewer donors than new patients, so the wait for a transplant continues to grow. To address this shortage, kidney paired donation (KPD) programs allow patients with living but biologically incompatible donors to exchange donors through cycles or chains initiated by altruistic (nondirected) donors, thereby increasing the supply of kidneys in the system. In many KPD programs a centralized algorithm determines which exchanges will take place to maximize the total number of transplants performed. This optimization problem has proven challenging both in theory, because it is NP-hard, and in practice, because the algorithms previously used were unable to optimally search over all long chains. We give two new algorithms that use integer programming to optimally solve this problem, one of which is inspired by the techniques used to solve the traveling salesman problem. These algorithms provide the tools needed to find optimal solutions in practice.

Stability of adversarial queues via fluid models

The subject of this paper is stability properties of adversarial queueing networks. Such queueing systems are used to model packet switch communication networks, in which packets are generated and routed dynamically, and have become a subject of research focus recently. Adversarial queueing networks are defined to be stable, if the number of packets stays bounded over time. A central question is determining which adversarial queueing networks are stable, when an arbitrary greedy packet routing policy is implemented. In this paper we show how stability of a queueing network can be determined by considering an associated fluid models. Our main result is that the stability of the fluid model implies the stability of an underlying adversarial queueing network. This opens an opportunity for analyzing stability of adversarial networks, using established stability methods from continuous time processes, for example, the method of Lyapunov function or trajectory decomposition. We demonstrate the use of these methods on several examples.

Limits of local algorithms over sparse random graphs

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-maximum independent sets in (sparse) random d-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞. Our result is based on an important clustering phenomena predicted first in the literature on spin glasses, and recently proved rigorously for a variety of constraint satisfaction problems on random graphs. Such properties suggest that the geometry of the solution space can be quite intricate. The specific clustering property, that we prove and apply in this paper shows that typically every two large independent sets in a random graph either have a significant intersection, or have a nearly empty intersection. As a result, large independent sets are clustered according to the proximity to each other. While the clustering property was postulated earlier as an obstruction for the success of local algorithms, such as for example, the Belief Propagation algorithm, our result is the first one where the clustering property is used to formally prove limits on local algorithms.

From Fluid Relaxations to Practical Algorithms for High-Multiplicity Job-Shop Scheduling: The Holding Cost Objective

We design an algorithm for the high-multiplicity job-shop scheduling problem with the objective of minimizing the total holding cost by appropriately rounding an optimal solution to a fluid relaxation in which we replace discrete jobs with the flow of a continuous fluid. The algorithm solves the fluid relaxation optimally and then aims to keep the schedule in the discrete network close to the schedule given by the fluid relaxation. If the number of jobs from each type grow linearly withN, then the algorithm is within an additive factorO( N) from the optimal (which scales asO( N2)); thus, it is asymptotically optimal. We report computational results on benchmark instances chosen from the OR library comparing the performance of the proposed algorithm and several commonly used heuristic methods. These results suggest that for problems of moderate to high multiplicity, the proposed algorithm outperforms these methods, and for very high multiplicity the overperformance is dramatic. For problems of low to moderate multiplicity, however, the relative errors of the heuristic methods are comparable to those of the proposed algorithm, and the best of these methods performs better overall than the proposed method.