A combinatorial property of flows on a cycle

TL;DR

This paper proves a combinatorial property of two-commodity flows on cycles, showing that one flow cannot uniformly decrease flow on all paths compared to another, and highlights its limitations for three or more commodities.

Contribution

It generalizes a known property from single-commodity to two-commodity flows on cycles and demonstrates its failure for three or more commodities.

Findings

Two-commodity flows on cycles satisfy a specific combinatorial property.

The property does not hold for three or more commodities, as shown by a counterexample.

The result extends understanding of flow behaviors in multi-commodity networks.

Abstract

In this paper, we prove a combinatorial property of flows on a cycle. $C(V,E)$ is an undirected cycle with two commodities: $\{s_{1},t_{1}\}, \{s_{2},t_{2}\}$;$r_1>0,r_2>0, \mathbf r=(r_i)_{i=1,2}$ and $f,f'$ are both feasible flows for $(C,(s_i,t_i)_{i=1,2},\mathbf r)$. Then $\exists i\in\{1,2\}, p\in P_i, f(p)>0, \forall e\in p, f(e)\geq f'(e)$ ; Here for each $i\in\{1,2\}$, let $P_i$ be the set of $s_i$-$t_i$ paths in $C$ and $P=\cup_{i=1,2}P_i$. This means given a two-commodity instance on a cycle, any two distinct network flow $f$ and $f'$, compared with $f'$, $f$ can't decrease every path's flow amount at the same time. This combinatorial property is a generalization from single-commodity case to two-commodity case, and we also give an instance to illustrate the combinatorial property doesn't hold on for $k-$commodity case when $k\geq 3$.