Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimumcost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasipolynomial time O(log²k/log log k)approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general DegreeBounded Directed Steiner Tree (DBDST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasipolynomial time (O(log n log k), O(log² n))bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first nontrivial result for the problem. While our costguarantee is nearly optimal, the degree violation factor of O(log²n) is an O(logmore »
Nodeweighted Network Design in Planar and Minorclosed Families of Graphs
We consider nodeweighted survivable network design (SNDP) in planar graphs and minorclosed families of graphs. The input consists of a nodeweighted undirected graph G = ( V , E ) and integer connectivity requirements r ( uv ) for each unordered pair of nodes uv . The goal is to find a minimum weighted subgraph H of G such that H contains r ( uv ) disjoint paths between u and v for each node pair uv . Three versions of the problem are edgeconnectivity SNDP (ECSNDP), elementconnectivity SNDP (ElemSNDP), and vertexconnectivity SNDP (VCSNDP), depending on whether the paths are required to be edge, element, or vertex disjoint, respectively. Our main result is an O ( k )approximation algorithm for ECSNDP and ElemSNDP when the input graph is planar or more generally if it belongs to a proper minorclosed family of graphs; here, k = max uv r ( uv ) is the maximum connectivity requirement. This improves upon the O ( k log n )approximation known for nodeweighted ECSNDP and ElemSNDP in general graphs [31]. We also obtain an O (1) approximation for nodeweighted VCSNDP when the connectivity requirements are in {0, 1, 2}; for higher connectivity our result more »
 Publication Date:
 NSFPAR ID:
 10293541
 Journal Name:
 ACM Transactions on Algorithms
 Volume:
 17
 Issue:
 2
 Page Range or eLocationID:
 1 to 25
 ISSN:
 15496325
 Sponsoring Org:
 National Science Foundation
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