Flow-cut gaps for integer and fractional multiflows

• Published in: Journal of combinatorial theory. Series B Vol. 103; no. 2; pp. 248 - 273
• Main Authors: Chekuri, Chandra, Shepherd, F. Bruce, Weibel, Christophe
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2013
• Subjects: Flow-cut gap; Integer flows; Multicommodity flows; Multiflows; Network flows; Outerplanar; Series-parallel; Multiflows; Outerplanar; Flow-cut gap; Series-parallel; Network flows; Integer flows; Multicommodity flows
• ISSN: 0095-8956; 1096-0902
• DOI: 10.1016/j.jctb.2012.11.002

Consider a routing problem instance consisting of a demand graphH=(V,E(H)) and a supply graphG=(V,E(G)). If the pair obeys the cut condition, then the flow-cut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C. It is well known that the flow-cut gap may be greater than 1 even in the case where G is the (series-parallel) graph K2,3. In this paper we are primarily interested in the “integer” flow-cut gap. What is the minimum value C such that there exists a feasible integer valued multiflow for H if each edge of G is given capacity C? We formulate a conjecture that states that the integer flow-cut gap is quantitatively related to the fractional flow-cut gap. In particular this strengthens the well-known conjecture that the flow-cut gap in planar and minor-free graphs is O(1) (Gupta et al. (2004) [13]) to suggest that the integer flow-cut gap is O(1). We give several technical tools and results on nontrivial special classes of graphs to give evidence for the conjecture and further explore the “primal” method for understanding flow-cut gaps; this is in contrast to and orthogonal to the highly successful metric embeddings approach. Our results include the following:•Let G be obtained by series-parallel operations starting from an edge st, and consider orienting all edges in G in the direction from s to t. A demand is compliant if its endpoints are joined by a directed path in the resulting oriented graph. We show that if the cut condition holds for a compliant instance and G+H is Eulerian, then an integral routing of H exists. This result includes as a special case, routing on a ring but is not a special case of the Okamura–Seymour theorem.•Using the above result, we show that the integer flow-cut gap in series-parallel graphs is 5. We also give an explicit class of routing instances that shows via elementary calculations that the flow-cut gap in series-parallel graphs is at least 2−o(1); this is motivated by and simplifies the proof by Lee and Raghavendra (2007) [15].•The integer flow-cut gap in k-Outerplanar graphs is cO(k) for some fixed constant c.•A simple proof that the flow-cut gap is O(logk⁎) where k⁎ is the size of a node-cover in H; this was previously shown by Günlük via a more intricate proof (Günlük (2007) [12]).