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...

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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 : : : : : : : : : : : : : : : : :

A new min-max theorem concerning bi-supermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and an earlier result of the first author on the minimum number of new directed edges whose addition makes a digraph k-edge-connected. As another consequence, we solve the corresponding node-connectivity augmentation problem in directed graphs.

In ad hoc wireless networks, it is crucial to minimize power consumption while maintaining key network properties. This work studies power assignments of wireless devices that minimize power while maintaining k-fault tolerance. Specifically, we require all links established by this power setting be symmetric and form a k-vertex connected subgraph of the network graph. This problem is known to be NP-hard. We show current heuristic approaches can use arbitrarily more power than the optimal solution. Hence, we seek approximation algorithms for this problem. We present three approximation algorithms. The first algorithm gives an O(kα)-approximation where α is the best approximation factor for the related problem in wired networks (the best α so far is O(log k).) With a more careful analysis, we show our second (slightly more complicated) algorithm is an O(k)-approximation. Our third algorithm assumes that the edge lengths of the network graph form a metric. In this case, we present simple and practical distributed algorithms for the cases of 2- and 3-connectivity with constant approximation factors. We generalize this algorithm to obtain an O(k 2c+2)-approximation for general k-connectivity (2 ≤ c ≤ 4 is the power attenuation exponent). Finally, we show that these approximation algorithms compare favorably with existing heuristics. We note that all algorithms presented in this paper can be used to minimize power while maintaining k-edge connectivity with guaranteed approximation factors.

: Let (N ) denote the weight of a minimum cut in an edge-weighted undirected network N , and n and m denote the numbers of vertices and edges, respectively. It is known that O(n 2k ) is an upper bound on the number of cuts with weights less than k(N ), where k 1 is a given constant. This paper rst shows that all cuts of weights less than k(N ) can be enumerated in O(m 2 n + n 2k m) time without using the maximum ow algorithm. The paper then proves for k < 4 3 that 0 n 2 is a tight upper bound on the number of cuts of weights less than k(N ), and that all those cuts can be enumerated in O(m 2 n+mn 2 log n) time. Keywords: minimum cuts, graphs, edge-splitting, polynomial algorithm Abbreviated title: Computing Small Cuts AMS subject classications: 05C35, 05C40 1 Introduction Let N stand for an undirected network with its edges being weighted by nonnegative real numbers. Counting the number of cuts with small weights, and deriving upper and lower bounds on their...

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We consider the problem of finding a smallest set of edges whose addition triconnects an undirected graph. This is a fundamental graph-theoretic problem that has applications in designing reliable networks and faulttolerant computing. We present a linear time sequential algorithm for the problem. This is a substantial improvement over the best previous algorithm for this problem, which runs in O(n(n+m)²) time on a graph with n vertices and m edges.

When k is even the min-max formula for the partition-constrained problem is a natural generalization of [3]. However this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge.