Improved approximation algorithms for degree-bounded network design problems with node connectivity requirements

Alina Ene, Ali Vakilian

Abstract: We consider degree bounded network design problems with element and vertex connectivity requirements. In the degree bounded Survivable Network Design (SNDP) problem, the input is an undirected graph G = ( V, E ) with weights w ( e ) on the edges and degree bounds b ( v ) on the vertices, and connectivity requirements r ( uv ) for each pair uv of vertices. The goal is to select a minimum-weight subgraph H of G that meets the connectivity requirements and it satisfies the degree bounds on the vertices: for each pair uv of vertices, H has r ( uv ) disjoint paths between u and v ; additionally, each vertex v is incident to at most b ( v ) edges in H . We give the first ( O (1), O (1) · b ( v )) bicriteria approximation algorithms for the degree-bounded SNDP problem with element connectivity requirements and for several degree-bounded SNDP problems with vertex connectivity requirements. Our algorithms construct a subgraph H whose weight is at most O (1) times the optimal such that each vertex v is incident to at most O (1) · b ( v ) edges in H . We can also extend our approach to network design problems in directed graphs with out-degree constraints to obtain ( O (1), O (1) · b + ( v )) bicriteria approximation.