Maximum edge-disjoint paths in planar graphs with congestion 2

• Published in: Mathematical programming Vol. 188; no. 1; pp. 295 - 317
• Main Authors: Séguin-Charbonneau, Loïc, Shepherd, F. Bruce
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2021; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Clustering; Combinatorics; Full Length Paper; Graphs; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Theoretical; Flows; Integrality gap; 90C27; 90C05; Edge-disjoint paths
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-020-01513-1

We study the maximum edge-disjoint path problem ( medp ) in planar graphs G = ( V , E ) with edge capacities u ( e ). We are given a set of terminal pairs s i t i , i = 1 , 2 … , k and wish to find a maximum routable subset of demands. That is, a subset of demands that can be connected by a family of paths that use each edge at most u ( e ) times. It is well-known that there is an integrality gap of Ω ( n ) for the natural LP relaxation, even in planar graphs (Garg–Vazirani–Yannakakis). We show that if every edge has capacity at least 2, then the integrality gap drops to a constant. This result is tight also in a complexity-theoretic sense: recent results of Chuzhoy–Kim–Nimavat show that it is unlikely that there is any polytime-solvable LP formulation for medp which has a constant integrality gap for planar graphs. Along the way, we introduce the concept of rooted clustering which we believe is of independent interest.