Network Design with Coverage Costs

We study network design with a cost structure motivated by redundancy in data traffic. We are given a graph, g groups of terminals, and a universe of data packets. Each group of terminals desires a subset of the packets from its respective source. The cost of routing traffic on any edge in the network is proportional to the total size of the distinct packets that the edge carries. Our goal is to find a minimum cost routing. We focus on two settings. In the first, the collection of packet sets desired by source-sink pairs is laminar. For this setting, we present a primal-dual based 2-approximation, improving upon a logarithmic approximation due to Barman and Chawla (2012). In the second setting, packet sets can have non-trivial intersection. We focus on the case where each packet is desired by either a single terminal group or by all of the groups, and the graph is unweighted. For this setting we present an O(log g)-approximation. Our approximation for the second setting is based on a novel spanner-type construction in unweighted graphs that, given a collection of g vertex subsets, finds a subgraph of cost only a constant factor more than the minimum spanning tree of the graph, such that every subset in the collection has a Steiner tree in the subgraph of cost at most O(log g) that of its minimum Steiner tree in the original graph. We call such a subgraph a group spanner.Comment: Updated version with additional result

Computing a Minimum-Cost kk-hop Steiner Tree in Tree-Like Metrics

We consider the problem of computing a Steiner tree of minimum cost under a $k$-hop constraint which requires the depth of the tree to be at most $k$. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time $n^{O(k)}$. For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if $k$ is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the $k$-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension

Further Improvements on Approximating the Uniform Cost-Distance Steiner Tree Problem

In this paper, we consider the Uniform Cost-Distance Steiner Tree Problem in metric spaces, a generalization of the well-known Steiner tree problem. Cost-distance Steiner trees minimize the sum of the total length and the weighted path lengths from a dedicated root to the other terminals, which have a weight to penalize the path length. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks, and the signal speed through the tree has to be considered besides the total length. Constant factor approximation algorithms for the uniform cost-distance Steiner tree problem have been known since the first mentioning of the problem by Meyerson, Munagala, and Plotkin. Recently, the approximation factor was improved from 2.87 to 2.39 by Khazraei and Held. We refine their approach further and reduce the approximation factor down to 2.15

Tighter Approximation for the Uniform Cost-Distance Steiner Tree Problem

Uniform cost-distance Steiner trees minimize the sum of the total length and weighted path lengths from a dedicated root to the other terminals. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks. They are a special case of general cost-distance Steiner trees, where different distance functions are used for total length and path lengths. We improve the best published approximation factor for the uniform cost-distance Steiner tree problem from 2.39 [Khazraei and Held, 2021] to 2.05. If we can approximate the minimum-length Steiner tree problem arbitrarily well, our algorithm achieves an approximation factor arbitrarily close to 1+1/?2. This bound is tight in the following sense. We also prove the gap 1+1/?2 between optimum solutions and the lower bound which we and all previous approximation algorithms for this problem use. Similarly to previous approaches, we start with an approximate minimum-length Steiner tree and split it into subtrees that are later re-connected. To improve the approximation factor, we split it into components more carefully, taking the cost structure into account, and we significantly enhance the analysis

Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics

We consider the problem of computing a Steiner tree of minimum cost under a k-hop constraint which requires the depth of the tree to be at most k. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time n^O(k). For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if k is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the k-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension and bounded doubling dimension