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.

💡 Research Summary

The paper investigates the integer flow‑cut gap for multiflow routing problems defined by a supply graph G and a demand graph H. The flow‑cut gap of an instance is the smallest uniform capacity C that must be assigned to every edge of G so that a feasible multiflow for H exists. While the fractional version of this gap has been extensively studied—most notably the conjecture that planar and minor‑free graphs have O(1) fractional gap—the integer version remains largely unexplored. The authors conjecture a quantitative relationship between the integer and fractional gaps, essentially proposing that the integer flow‑cut gap is also bounded by a constant in the same families of graphs.

The first major contribution focuses on series‑parallel graphs. Starting from a single edge st, the graph G is built using series and parallel operations. All edges are oriented from s to t, and a demand is called “compliant” if its two endpoints are connected by a directed path in this orientation. The authors prove that if the cut condition holds for a compliant instance and the combined graph G + H is Eulerian (every vertex has even degree), then there exists an integral routing of H. This result is obtained via a “primal” method: instead of constructing a dual certificate, the proof directly builds a set of edge‑disjoint directed paths that satisfy all demands, showing that unit capacity (C = 1) suffices under the stated structural conditions.

The second contribution establishes a tight bound for the integer flow‑cut gap in all series‑parallel graphs. They show that a uniform capacity of 5 on every edge of G guarantees an integral multiflow whenever the cut condition is satisfied, i.e., the integer flow‑cut gap is exactly 5. To complement this upper bound, they present a simple family of instances that achieve a lower bound of 2 − o(1), thereby simplifying the earlier proof by Lee and Raghavendra that the (fractional) gap in series‑parallel graphs is at least 2. This demonstrates that the integer gap can indeed exceed 1, and that the constant 5 is essentially optimal for the class.

The third result extends the analysis to k‑outerplanar graphs. By exploiting the layered structure of outerplanarity, the authors prove that the integer flow‑cut gap grows at most exponentially in k, specifically as c^{O(k)} for some absolute constant c. Although the bound becomes large for deep outerplanarity, it remains polynomial for any fixed k, implying that integral routings are still efficiently obtainable in these graphs.

Finally, the paper revisits the relationship between the gap and the size k* of a minimum node‑cover in the demand graph H. Building on a prior result by Günlük, they give a concise primal‑based proof that the integer flow‑cut gap is O(log k*). Their argument partitions the demand graph according to the covering vertices, routes each part with a small constant capacity, and then combines the routings, yielding a logarithmic dependence on k*.

Overall, the work provides several non‑trivial upper bounds on the integer flow‑cut gap for important graph families, introduces a versatile primal technique for constructing integral routings, and strengthens the connection between integer and fractional gaps. These contributions not only advance the theoretical understanding of multiflow routing but also suggest that the long‑standing conjecture of an O(1) flow‑cut gap in planar and minor‑free graphs may extend to the integral setting as well.