Tight approximation algorithm for connectivity augmentation problems

Abstract

The S-connectivity λ G S ( u , v ) of ( u , v ) in a graph G is the maximum number of uv-paths that no two of them have an edge or a node in S − { u , v } in common. The corresponding Connectivity Augmentation ( CA ) problem is: given a graph G 0 = ( V , E 0 ) , S ⊆ V , and requirements r ( u , v ) on V × V , find a minimum size set F of new edges (any edge is allowed) so that λ G 0 + F S ( u , v ) ⩾ r ( u , v ) for all u , v ∈ V . Extensively studied particular choices of S are the edge- CA (when S = ∅ ) and the node- CA (when S = V ). A. Frank gave a polynomial algorithm for undirected edge- CA and observed that the directed case even with rooted { 0 , 1 } -requirements is at least as hard as the Set-Cover problem (in rooted requirements there is s ∈ V − S so that if r ( u , v ) > 0 then: u = s for directed graphs, and u = s or v = s for undirected graphs). Both directed and undirected node- CA have approximation threshold Ω ( 2 log 1 − ε n ) . The only polylogarithmic approximation ratio known for CA was for rooted requirements— O ( log n ⋅ log r max ) = O ( log 2 n ) , where r max = max u , v ∈ V r ( u , v ) . No nontrivial approximation algorithms were known for directed CA even for r ( u , v ) ∈ { 0 , 1 } , nor for undirected CA with S arbitrary. We give an approximation algorithm for the general case that matches the known approximation thresholds. For both directed and undirected CA with arbitrary requirements our approximation ratio is: O ( log n ) for S ≠ V arbitrary, and O ( r max ⋅ log n ) for S = V .