Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous Õ(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in �� � with n 2 processors; this gives the first proof that the minimum cut problem can be solved in ���. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of � of the minimum cut’s in expected Õ(n 2 � ) time, or in �� � with n 2 � processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected Õ(n 2(r�1) ) time, or in �� � with n 2(r�1) processors. The “trace ” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semi-duality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log 3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n 2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n 2 log 3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner. 1 Introduction The minimum cut problem has been studied for many years as a fundamental graph optimization problem with numerous applications. Initially, th...

The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is ]P-complete, and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and 1=ffl an estimate for the failure probability that is accurate to within a relative error of 1 \Sigma ffl with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of k-connectivity, strong connectivity in directed Eulerian graph...

We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)-processor NC algorithm for finding a (2 + ffl)-approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)-approximation to the minimum cut. This reduction involves a natural combinatorial Set-Isolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the set-isolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum k-way cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...

Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worst-case time bounds for the problem. We conduct experimental evaluation the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem.

We present new algorithms, based on random sampling, that find maximum flows in undirected uncapacitated graphs. Our algorithms dominate augmenting paths over all parameter values (number of vertices and edges and flow value). They also dominate blocking flows over a large range of parameter values. Furthermore, they achieve time bounds on graphs with parallel (equivalently, capacitated) edges that previously could only be achieved on graphs without them. The key contribution of this paper is to demonstrate that such an improvement is possible. This shows that augmenting paths and blocking flows are non-optimal, and reopens the question of how fast we can find a maximum flow. We improve known time bounds by only a small (but polynomial) factor, and the complicated nature of our algorithms suggests they will not be practical. A new idea of our algorithm is to find flow by diminishing cuts instead of augmenting paths. Rather than finding a way to push flow from the source to the sink, we...

We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)-edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)-time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)-times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.

In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to re-emphasize the importance and utility of this algorithm.

Can we automatically compose a large set of Wiktionaries and translation dictionaries to yield a massive, multilingual dictionary whose coverage is substantially greater than that of any of its constituent dictionaries? The composition of multiple translation dictionaries leads to a transitive inference problem: if word A translates to word B which in turn translates to word C, what is the probability that C is a translation of A? The paper introduces a novel algorithm that solves this problem for 10,000,000 words in more than 1,000 languages. The algorithm yields PANDIC-TIONARY, a novel multilingual dictionary. PANDICTIONARY contains more than four times as many translations than in the largest Wiktionary at precision 0.90 and over 200,000,000 pairwise translations in over 200,000 language pairs at precision 0.8.

Randomization has become a pervasive technique in combinatorial optimization. We survey our thesis and subsequent work, which uses four common randomization techniques to attack numerous optimization problems on undirected graphs. 1 Introduction Randomization has become a pervasive technique in combinatorial optimization. Randomization has been used to develop algorithms that are faster, simpler, and/or better-performing than previous deterministic algorithms. This article surveys our thesis [Kar94], which presents randomized algorithms for numerous problems on undirected graphs. Our work uses four important randomization techniques: Random Selection, which lets us easily choose a "typical" element of a set, avoiding rare "bad" elements; Random Sampling, which provides a quick way to build a small, representative subproblem of a larger problem for quick analysis; Randomized Rounding, which lets us transform fractional problem solutions into integral ones; and Monte Carlo Simulatio...