We consider a new problem of designing a network with small @math - @math effective resistance. In this problem, we are given an undirected graph @math , two designated vertices @math , and a budget @math . The goal is to choose a subgraph of @math with at most @math edges to minimize the @math - @math effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.
Other combinatorial connectivity requirements were also considered. A natural variation is to require @math internally vertex disjoint paths for every pair of vertices @math . This problem is much harder to approximate @cite_10 @cite_2 , but there are good approximation algorithms for global connectivity @cite_22 @cite_17 and when the maximum connectivity requirement is small @cite_39 @cite_36 . Another natural problem is to require a path of length @math between every pair of vertices @math . This problem is also hard to approximate in general but there are better approximation algorithms when every edge has the same cost and the same length @cite_23 .
@cite_22 @cite_36 @cite_39 @cite_23 @cite_2 @cite_10 @cite_17
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We study undirected networks with edge costs that satisfy the triangle inequality. Let @math denote the number of nodes. We present an @math -approximation algorithm for a generalization of the metric-cost subset @math -node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each @math , there exists a @math -node connected graph whose cost is within a factor of @math of the cost of any simple @math -edge connected graph. Based on our @math -approximation algorithm, we present an @math -approximation algorithm for the metric-cost node-connectivity survivable network design problem, where @math denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704-720] recently proved, assuming NP @math DTIME( @math ), a hardness-of-approximation lower bound of @math for the subset @math -node-connectivity problem, where @math denotes a small positive number. 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 give the first strong lower bound on the approximability of SNDP, showing that the problem admits no efficient @math ratio approximation for any fixed @math , unless @math . We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . A typical problem in network design is to find a minimum-cost sub-network H of a given network G such that H satisfies some prespecified connectivity requirements. Our focus is on approximation algorithms for designing networks that satisfy vertex connectivity requirements. Our main tool is a linear programming relaxation of the following setpair formulation due to Frank and Jordan: a setpair consists of two subsets of vertices (of the given network G); each setpair has an integer requirement, and the goal is to find a minimum-cost subset of the edges of G sucht hat each setpair is covered by at least as many edges as its requirement. We introduce the notion of skew bisupermodular functions and use it to prove that the basic solutions of the linear program are characterized by “non-crossing families” of setpairs. This allows us to apply Jain’s iterative rounding method to find approximately optimal integer solutions. We give two applications. (1) In the k-vertex connectivity problem we are given a (directed or undirected) graph G=(V,E) with non-negative edge costs, and the task is to find a minimum-cost spanning subgraph H such that H is k-vertex connected. Let n=|V|, and let e<1 be a positive number such that k≤(1−e)n. We give an @math -approximation algorithm for both problems (directed or undirected), improving on the previous best approximation guarantees for k in the range @math . (2)We give a 2-approximation algorithm for the element connectivity problem, matching the previous best approximation guarantee due to Fleischer, Jain and Williamson.