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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).

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.

We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimum-cost set of edges such that there are r ij vertex-disjoint paths between vertices i and j. In the case for which r ij 2 f0; 1; 2g for all i; j, we can find a solution of cost no more than 3 times the optimal cost in polynomial time. In the case in which r ij = k for all i; j, we can find a solution of cost no more than 2H(k) times optimal, where H(n) = 1 + 1 2 + \Delta \Delta \Delta + 1 n . No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems. 1 Introduction Let G = (V; E) be an undirected graph with non-negative costs c e 0 on all edges e 2 E. In...

In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them.

We consider the problem of deploying or repairing a sensor network to guarantee a specified level of multi-path connectivity (k-connectivity) between all nodes. Such a guarantee simultaneously provides fault tolerance against node failures and high overall network capacity (by the max-flow min-cut theorem). We design and analyze the first algorithms that place an almostminimum number of additional sensors to augment an existing network into a k-connected network, for any desired parameter k. Our algorithms have provable guarantees on the quality of the solution. Specifically, we prove that the number of additional sensors is within a constant factor of the absolute minimum, for any fixed k. We have implemented greedy and distributed versions of this algorithm, and demonstrate in simulation that they produce high-quality placements for the additional sensors.

Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy four-connectivity. A four-connected planar graph has no separating triangles, i.e. cycles of length 3 which are not a face. We show

Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present survey-type paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization.

Generalizing and unifying earlier results of W. Mader and of the second and third authors, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satisfies local edgeconnectivity prescriptions. An extension of Edmonds' theorem on disjoint arborescences is also deduced along with a new sufficient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems. 1. INTRODUCTION AND PRELIMINARIES Our main concern, the edge-connectivity augmentation problem, is as follows. What is the minimum number (or, more generally, the minimum cost) fl of new edges to be added to M so that in the resulting graph M 0 the local edge-connectivity (x; y; M 0 ) between every pair of nodes x; y is at least a prescribed value r(x; y)? Several ...