Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs

We present a 6-approximation algorithm for the minimum-cost $k$-node connected spanning subgraph problem, assuming that the number of nodes is at least $k^3(k-1)+k$. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for $k$-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 $k$.Comment: 20 pages, 1 figure, 28 reference

Single-Sink Network Design with Vertex Connectivity Requirements

We study single-sink network design problems in undirected graphs with vertex connectivity requirements. The input to these problems is an edge-weighted undirected graph $G=(V,E)$, a sink/root vertex $r$, a set of terminals $T subseteq V$, and integer $k$. The goal is to connect each terminal $t in T$ to $r$ via $k$ emph{vertex-disjoint} paths. In the {em connectivity} problem, the objective is to find a min-cost subgraph of $G$ that contains the desired paths. There is a $2$-approximation for this problem when $k le 2$ cite{FleischerJW} but for $k ge 3$, the first non-trivial approximation was obtained in the recent work of Chakraborty, Chuzhoy and Khanna cite{ChakCK08}; they describe and analyze an algorithm with an approximation ratio of $O(k^{O(k^2)}log^4 n)$ where $n=|V|$. In this paper, inspired by the results and ideas in cite{ChakCK08}, we show an $O(k^{O(k)}log |T|)$-approximation bound for a simple greedy algorithm. Our analysis is based on the dual of a natural linear program and is of independent technical interest. We use the insights from this analysis to obtain an $O(k^{O(k)}log |T|)$-approximation for the more general single-sink {em rent-or-buy} network design problem with vertex connectivity requirements. We further extend the ideas to obtain a poly-logarithmic approximation for the single-sink {em buy-at-bulk} problem when $k=2$ and the number of cable-types is a fixed constant; we believe that this should extend to any fixed $k$. We also show that for the non-uniform buy-at-bulk problem, for each fixed $k$, a small variant of a simple algorithm suggested by Charikar and Kargiazova cite{CharikarK05} for the case of $k=1$ gives an $2^{O(sqrt{log |T|})}$ approximation for larger $k$. These results show that for each of these problems, simple and natural algorithms that have been developed for $k=1$ have good performance for small $k > 1$

Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs

In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G = (V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k > 0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ? 3. In this paper, we present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k ? 3, that runs in polynomial-time Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem

Approximation Algorithms for (S,T)-Connectivity Problems

We study a directed network design problem called the $k$-$(S,T)$-connectivity problem; we design and analyze approximation algorithms and give hardness results. For each positive integer $k$, the minimum cost $k$-vertex connected spanning subgraph problem is a special case of the $k$-$(S,T)$-connectivity problem. We defer precise statements of the problem and of our results to the introduction. For $k=1$, we call the problem the $(S,T)$-connectivity problem. We study three variants of the problem: the standard $(S,T)$-connectivity problem, the relaxed $(S,T)$-connectivity problem, and the unrestricted $(S,T)$-connectivity problem. We give hardness results for these three variants. We design a $2$-approximation algorithm for the standard $(S,T)$-connectivity problem. We design tight approximation algorithms for the relaxed $(S,T)$-connectivity problem and one of its special cases. For any $k$, we give an $O(\log k\log n)$-approximation algorithm, where $n$ denotes the number of vertices. The approximation guarantee almost matches the best approximation guarantee known for the minimum cost $k$-vertex connected spanning subgraph problem which is $O(\log k\log\frac{n}{n-k})$ due to Nutov in 2009

The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04)