Flow-Cut Gaps for Integer and Fractional Multiflows

Consider a routing problem consisting of a demand graph H and a supply graph G. If the pair obeys the cut condition, then the flow-cut gap for this instance is the minimum value C such that there is a feasible multiflow for H if each edge of G is given capacity C. The flow-cut gap can be greater than 1 even when G is the (series-parallel) graph K_{2,3}. In this paper we are primarily interested in the “integer” flow-cut gap. What is the minimum value C such that there is a feasible integer valued multiflow for H if each edge of G is given capacity C? We conjecture that the integer flow-cut gap is quantitatively related to the fractional flow-cut gap. This strengthens the well-known conjecture that the flow-cut gap in planar and minor-free graphs is O(1) to suggest that the integer flow-cut gap is O(1). We give several results on non-trivial special classes of graphs supporting this conjecture and further explore the “primal” method for understanding flow-cut gaps. Our results include:

• 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. If the cut condition holds for a compliant instance and G+H is Eulerian, then an integral routing of H exists.
• The integer flow-cut gap in series-parallel graphs is 5. We also give an explicit class of instances that shows via elementary calculations that the flow-cut gap in series-parallel graphs is at least 2-o(1); this simplifies the proof by Lee and Raghavendra.
• The integer flow-cut gap in k-Outerplanar graphs is c^{O(k)} for some fixed constant c.
• A simple proof that the flow-cut gap is O(\log k^) where k^ is the size of a node-cover in H; this was previously shown by G"unl"uk via a more intricate proof.