Abstract: An efficient heuristic is presented for the problem of finding a minimum-size k- connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimum-size k-node connected spanning subgraph of an undirected graph 1+[1=k], minimum-size k-node connected spanning subgraph of a directed graph 1+[1=k], minimum-size k-edge connected spanning subgraph of an undirected graph 1+[2=(k + 1)], and minimum-size k-edge connected spanning subgraph of a directed graph 1+[4= p k].

Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.

Introduction The maximum flow problem is a classical optimization problem with many applications; see e.g. [1, 18, 39]. Algorithms for this problem have been studied for over four decades. Recently, significant improvements have been made in theoretical performance of maximum flow algorithms. In this survey we put these results in perspective and provide pointers to the literature. We assume that the reader is familiar with basic flow algorithms, including Dinitz' blocking flow algorithm [13]. 2 Preliminaries The maximum flow problem is to find a flow of the maximum value given a graph G with arc capacities, a source s, and a sink t, Here a flow is a function on arcs that satisfies capacity constraints for all arcs and conservation constraints for all vertices except the source and the sink. For more details, see [1, 18, 39]. We distinguish between directed

Consider the minimum size k-edge-connected spanning subgraph problem: given a positive integer k and a k-edge-connected graph G, nd a k-edge-connected spanning subgraph of G with the minimum number of edges. This problem is known to be NP-complete. Khuller and Raghavachari presented the rst algorithm which, for all k, achieves a performance ratio smaller than a constant which is less than two. They proved an upper bound of 1.85 for the performance ratio of their algorithm. Currently, the best known performance ratio for the problem is 1+2=(k+ 1), achieved by a slower algorithm of Cheriyan and Thurimella. In this paper, we improve Khuller and Raghavachari's analysis, proving that the performance ratio of their algorithm is smaller than 1.7 for large enough k, and that it is at most 1.75 for all k. Second, we show that the minimum size 2-edge-connected spanning subgraph problem is MAX SNP-hard. A preliminary version of this paper appeared in the Proceedings of the 8 th Annual ACM...

A k-separator (k-shredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be k-node connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of k-separators is #P-complete. However, we present an O(k )-time (deterministic) algorithm for finding all the k-shredders. This solves an open question: efficiently find a k-separator whose removal maximizes the number of connected 4, our running time is within a factor of k of the fastest algorithm known for testing k-node connectivity. One application of shredders is in increasing the node connectivity from k to (k +1)by effi tly adding an (approximately) minimum number of new edges. Jord'an [JCT(B) 1995] gaveanO(n )-time augmentation algorithm such that the number of new edges is within an additive term of (k 2) from a lower bound. We improve the running time to ), while achieving the same performance guarantee. For k 4, the running time compares favorably with the running time for testing k-node connectivity.

We describe an O(min(m; n 3=2 )m 1=2 )-time algorithm for finding maximum flows in undirected networks with unit capacities and no parallel edges. This improves upon the previous bound of Karzanov and Even and Tarjan when m = !(n 3=2 ), and upon a randomized bound of Karger when v = \Omega\Gamma n 7=4 =m 1=2 ). (Here v is the maximum flow value.) 1 Introduction In this paper we consider the undirected maximum flow problem in a network with unit capacities and no parallel edges. Until recently, the fastest known way to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov [7] and Even and Tarjan [2] have shown that Dinitz's blocking flow algorithm [1], applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. (Here n and m is the number of input vertices and edges, respectively.) Recently, Karger [6] developed two randomized algorithms for the undirected problem, with running times of O (m 5...

A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes k-path queries in output-sensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.

Abstract The (vertex) connectivity ^ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known algorithm for finding ^. For a digraph with n vertices, m edges and connectivity ^ the time bound is O((n + minf^5=2; ^n3=4g)m). This improves the previous best bound of O((n + minf^3; ^ng)m). For an undirected graph both of these bounds hold with m replaced by ^n. Our approach uses expander graphs to exploit nesting properties of certain separation triples. 1 Introduction The (vertex) connectivity ^ of a graph is the smallest number of vertices whose deletion separates or trivializes the graph. (Other basic terminology is defined at the end of this section.) This is a central concept of graph theory [11]. Computing the connectivity is posed as Research Problem 5.30 in [1] where in fact a linear-time algorithm is conjectured. Yet relatively little progress has been made on computing connectivity.1 If the conjectured linear-time algorithm exists it involves techniques that are radically different from the known ones.

Karger, Motwani and Ramkumar have shown that there is no constant approx-imation algorithm to find a longest cycle in a Hamiltonian graph, and they conjectured this is the case even for graphs with bounded degree. On the other hand,Feder, Motwani and Subi have shown that there is a polynomial time algorithm for finding a cycle of length nlog3 2 in a 3-connected cubic n-vertex graph. In this paper,we show that if G is a 3-connected n-vertex graph with maximum degree at most d, then one can find, in O(n3) time, a cycle in G of length at least \Omega (nlogb 2), where b = 2(d- 1)² + 1.

This paper is organized as follows. In section 2 we introduce notation, concepts and definitions. In section 3 we state three formulations of the GH tree and give a high-level description of an algorithm for which n=2 calls to the min cut oracle are sufficient to discover the tree. In section 4 we state a Ramsey-type theorem about networks which is of independent interest, together with an open problem. Due to space limitations, we try to give the flavor of our approach, rather than precise claims and proofs. 2 Notation A network consists of n vertices and m undirected edges. Vertices are denoted v a ; v i ; v 1 , while capital letters I; X; S; Y represent sets of vertices. An edge between vertices v i ; v j is denoted (i; j). Lower case c is used to denote connection weight, e.g. c(i; j) is the capacity of the edge (i; j), while upper case C is used in various ways to denote a cut or its capacity. For example,