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 ...

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 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).

A cactus tree is a simple data structure that represents all minimum cuts of a weighted graph in linear space. We describe the first algorithm that can build a cactus tree from the asymptotically fastest deterministic algorithm that finds all minimum cuts in a weighted graph --- the Hao-Orlin minimum cut algorithm. This improves the time to construct the cactus in graphs with n vertices and m edges from O(n 3 ) to O(nm log n 2 =m).

Abstract not found

This paper presents a deterministic O(nm log n + n 2 log 2 n) = ~ O(nm) time algorithm for splitting o all edges incident to a vertex s of even degree in a multigraph G, where n and m are the numbers of vertices and links (= vertex pairs between which G has an edge) in G, respectively. Based on this, many graph algorithms using edge-splitting can run faster. For example, the edge-connectivity augmentation problem in an undirected multigraph can be solved in ~ O(nm) time, which is an improvement over the previously known randomized ~ O(n 3 ) bound and deterministic ~ O(n 2 m) bound. 1 Introduction Let G = (V; E) stand for an undirected multigraph with a set V of vertices and a set E of edges, where an edge with end vertices u and v is denoted by (u; v). A singleton set fxg may be simply written as x, and \ " implies proper inclusion while \ " means \ " or \ = ". For two disjoint subsets X;Y V , we denote by EG (X; Y ) the set of edges, one of whose end vertices is i...

The Traveling Salesman Problem (TSP) is a benchmark problem in combinatorial optimization. It was one of the very first problems used for developing and testing approaches to solving large integer programs, including cutting plane algorithms and branch-and-cut algorithms. Much of the research in this area has been focused on finding new classes of facets for the TSP polytope, and much less attention has been paid to algorithms for separating from these classes of facets. In this paper, we consider the problem of finding violated comb inequalities. If there are no violated subtour constraints in a fractional solution of the TSP, a comb inequality may not be violated by more than 1. Given a fractional solution in the subtour elimination polytope whose graph is planar, we either find a violated comb inequality or determine that there are no comb inequalities violated by 1. Our algorithm runs in O(n + MC(n)) time, where MC(n) is the time to compute a cactus representation of all minimum cu...

Two vertices of an undirected graph are called k-edge-connected if there exist k edge-disjoint paths between them (equivalently: they cannot be disconnected by the removal of less than k edges from the graph). Equivalence classes of this relation are called classes of k-edge-connectivity, or k-edge-connected components. This paper describes graph structures relevant to classes of 4-edge-connectivity and traces their transformations as new edges are inserted into the graph. Data structures and an algorithm to maintain incrementally these classes are given. Starting with the empty graph, any sequence of q Same-4-Class? queries and n Insert-Vertex and m Insert-Edge updates can be performed in O(q + m + n log n) total time. Each individual query requires O(1) time in the worst-case. In addition, an algorithm for maintaining the classes of k-edge-connectivity (k arbitrary) in a (k \Gamma 1)-edge-connected graph is presented. Its complexity is O(q+m+n), with n log(n=k)) preprocessing, where M is the number of edges initially in the graph and n is the number of its vertices.

For a given undirected graph G = (V; E; c G ) with edges weighted by nonnegative reals c G : E ! R + , let G (k) stand for the minimum amount of weights which needs to be added to make G k-edge-connected, and G 3 (k) be the resulting graph obtained from G. This paper rst shows that function G over the entire range k 2 [0; +1] can be computed in O(nm + n 2 log n) time, and then shows that all G 3 (k) in the entire range can be obtained from O(n log n) weighted cycles, and such cycles can be computed in O(nm+n 2 log n) time, where n and m are the numbers of vertices and edges, respectively. 1 Introduction Let G = (V; E; c G ) be an edge-weighted undirected graph with a set V of vertices, a set E of edges, and a weight function c G : E !R + , where R + denotes the set of nonnegative reals. We denote n = jV j and m = jEj. An edge with end vertices u and v is denoted by (u; v). A singleton set fxg may be simply written as x, and \ " implies proper inclusion while \ " ...