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 present an O(log n• log k)-approximation algorithm for the problem of finding k-vertex connected spanning subgraph of minimum cost, where n is the number of vertices in the input graph, and k is the connectivity requirement. Our algorithm works for both directed and undirected graphs. The best known approximation guarantees for these problems are O(ln k• min √k,n n-k ln k ) by Kortsarz and Nutov, and O(ln k ) in the case of undirected graphs where n≥ 6k2 by Cheriyan, Vempala, and Vetta. Our algorithm is the first that has a polylogarithmic guarantee for all values of k. Combining our algorithm with the algorithm of Kortsarz and Nutov in case of small k, e.g., k We study the following network design problem: Given a communication network, find a minimum cost subset of missing links such that adding these links to the network makes every pair of points within distance at most d from each other. The problem has been studied earlier [17] under the assumption that all link costs as well as link lengths are identical, and was shown to be R(logn)-hard for every d 2 4. We present a novel linear programming based approach to obtain an O(log la log d) approximation algorithm for the case of uniform link lengths and costs. We also extend the Cl(Iogn) hardness to d E Z, 3). On the other hand, if link costs can vary, we show that the prob” ‘-’ n lem is n(Z s )hard for d > 3. This version of our problem can be viewed as a special case of the minimum cost d-spanner problem and thus our hardness result applies there as well. For d = 2, however, we show that the problem continues to be O(logn) approximable by giving an O(log n)-approximation to the more general minimum cost Z-spanner problem. An n(2”s’-’ “)-hardness result also holds when all link costs are identical but link lengths may vary (applies even when all lengths are 1 or 2). Our reduction from the label cower problem [3] also applies to another well-studied network design problem. We show that the directed genemlized steiner network problem [6] is n(2 I’&-’ “)-hard, significantly improving upon the Q(logn) hardness known prior to our work. We also present O(n log d) approximation algorithm for our problem under arbitrary link costs and polynomially bounded link lengths. Same result holds for the minimum cost d-spanner problem. Finally, all our positive results extend to the case where each pair (u,u) of nodes has a distinct distance requirement, say d(u, v). The approximation guarantees above hold provided d is replaced by max,,, d(u, v). All our algorithmic as well as hardness results hold for both undirected and directed versions of the problem. Sanjeev Khanna Optimizing parameters of Two-Prover-One-Round Game (2P1R) is an important task in PCPs literature as it would imply a smaller PCP with the same or stronger soundness. While this is a basic question in PCPs community, the connection between the parameters of PCPs and hardness of approximations is sometimes obscure to approximation algorithm community. In this paper, we investigate the connection between the parameters of 2P1R and the hardness of approximating the class of so-called connectivity problems, which includes as subclasses the survivable network design and (multi)cut problems. Based on recent development on 2P1R by Chan (STOC 2013) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form kσ, for some (very) small constant σ > 0, to hardness results of the form kc for some explicit constant c, where k is a connectivity parameter. In addition, we show how to convert these hardness into hardness results of the form Dc', where D is the number of demand pairs (or the number of terminals). 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 a 6-approximation algorithm for the minimum-cost @math -node connected spanning subgraph problem, assuming that the number of nodes is at least @math . We apply a combinatorial preprocessing, based on the Frank--Tardos algorithm for @math -outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of @math .