A Primal-Dual Algorithm for the Generalized Prize-Collecting Steiner Forest Problem

2017 ◽  
Vol 5 (2) ◽  
pp. 219-231
Author(s):  
Lu Han ◽  
Da-Chuan Xu ◽  
Dong-Lei Du ◽  
Chen-Chen Wu
Keyword(s):  
Dual Algorithm ◽  
Steiner Forest ◽  
Primal Dual ◽  
Prize Collecting

scholarly journals Node-weighted Network Design in Planar and Minor-closed Families of Graphs

2021 ◽  
Vol 17 (2) ◽  
pp. 1-25
Author(s):  
Chandra Chekuri ◽  
Alina Ene ◽  
Ali Vakilian
Keyword(s):  
Network Design ◽  
Planar Graphs ◽  
Constant Factor ◽  
Dual Algorithm ◽  
Vertex Connectivity ◽  
Logarithmic Factor ◽  
Steiner Forest ◽  
Primal Dual ◽  
Vertex Disjoint ◽  
General Graphs

We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted 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 edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP), and vertex-connectivity SNDP (VC-SNDP), 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 EC-SNDP and Elem-SNDP when the input graph is planar or more generally if it belongs to a proper minor-closed 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 node-weighted EC-SNDP and Elem-SNDP in general graphs [31]. We also obtain an O (1) approximation for node-weighted VC-SNDP when the connectivity requirements are in {0, 1, 2}; for higher connectivity our result for Elem-SNDP can be used in a black-box fashion to obtain a logarithmic factor improvement over currently known general graph results. Our results are inspired by, and generalize, the work of Demaine, Hajiaghayi, and Klein [13], who obtained constant factor approximations for node-weighted Steiner tree and Steiner forest problems in planar graphs and proper minor-closed families of graphs via a primal-dual algorithm.

Study of a primal-dual algorithm for equality constrained minimization

2014 ◽  
Vol 59 (3) ◽  
pp. 405-433 ◽  
Author(s):  
Paul Armand ◽  
Joël Benoist ◽  
Riadh Omheni ◽  
Vincent Pateloup
Keyword(s):  
Constrained Minimization ◽  
Dual Algorithm ◽  
Primal Dual
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