We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown boundeddegree graph is connected, k-connected (for k ? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is ffl-away from having the property. For example, the 2-Connectivity algorithm rejects (w.h.p.) any N-vertex d-degree graph for which more ...

We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.

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

A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP -hard. We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2-vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP -hard as well. We also consider the case where the graph has edge weigh...

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...

We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to 1 � O(�log n)/k).

We first consider the problem of partitioning the edges of a graph G into bipartite cliques such that the total order of the cliques is minimized, where the order of a clique is the number of vertices in it. It is shown that the problem is NP-complete. We then prove the existence of a partition of small total order in a sufficiently dense graph and devise an efficient algorithm to compute such a partition. It turns out that our algorithm exhibits a trade-off between the total order of the partition and the running time. Next, we define the notion of a compression of a graph G and use the result on graph partitioning to efficiently compute an optimal compression for graphs of a given size. An interesting application of the graph compression result arises from the fact that several graph algorithms can be adapted to work with the compressed representation of the input graph, thereby improving the bound on their running times, particularly on dense graphs. This makes use of the trade-off ...

Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Edge-Connectivity Problems 3 2.1 Weighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.1 2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 Vertex-Connectivity Problems 11 3.1 Weighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.2 Unweighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.2.1 2 Vertex-Connectivity : : : : : : : : : : : : : : : : :

In this paper we evaluate a method of using interleaved spanning trees to compose a resilient, high performance overlay mesh. Though spanning trees of arbitrary type could be used to construct an overlay mesh, we focus on a distributed algorithm that computes k minimum spanning trees on an arbitrary graph. The principal motivation behind this strategy is to provide applications with a k-redundant, high quality mesh suitable for demanding applications like A/V broadcast, video conferencing, data collection, multi-path routing, and file mirroring/transfer. We elaborate details of k-MST, pointing out advantages and potential problem points of the protocol, and then analyze its performance using a variety of metrics with simulation as well as a functional PlanetLab implementation.

Abstract — We consider the problem of broadcasting a live stream of data in an unstructured network. The broadcasting problem has been studied extensively for edge-capacitated networks. We give the first proof that whenever demand λ + ε is feasible for ε>0, a simple local-control algorithm is stable under demand λ, and as a corollary a famous theorem of Edmonds. We then study the node-capacitated case and show a similar optimality result for the complete graph. We study through simulation the delay that users must wait in order to playback a video stream with a small number of skipped packets, and discuss the suitability of our algorithms for live video streaming. I.