Srinwanti Debgupta, Martin Skutella, Philipp Warode
原文摘要
In contrast to the max-flow/min-cut theorem for single-commodity flows, the cut condition is necessary but not sufficient for the existence of directed multiflows, even with only two commodities. Poremba and Shepherd (IPCO 2023) conjecture that the cut condition is sufficient for all capacities and demands on a directed two-commodity topology if and only if it contains neither of two specific topologies as a relevant minor. We prove this conjecture. As a consequence, the fractional and integral flow-cut gaps of every directed two-commodity topology coincide and are either 1 or 2. We then investigate stronger necessary conditions for multiflow feasibility and their implications for integrality. We introduce a hierarchy of five conditions, ranging from the classical cut condition through conditions based on blocking edge sets and binary edge lengths to the linear programming dual characterization of fractional feasibility. The corresponding classes of sufficient topologies are strictly nested. We extend the Nagamochi-Ibaraki theorem: whenever one of the first four conditions is sufficient on a topology, integral capacities and demands satisfying that condition admit an integral multiflow, computable in polynomial time. Finally, we prove that recognizing topologies on which any of these four conditions is sufficient is coNP-complete, even for two commodities.
We study optimal contracts when an agent allocates continuous effort subject to polymatroid capacity constraints. Effort produces a linear expected reward and incurs a cost. With binary outcomes and limited liability, the principal chooses a single reward share. For linear costs, an optimal contract can be computed in polynomial time. For quadratic costs, the agent has a unique best response computable by polynomially many submodular minimizations, yet optimizing the principal's share is NP-hard even for explicitly represented cut-based rank functions. With $n$ activities, the best-response path can have $2^{n-O(\log n)}$ breakpoints even when each reward coefficient and each rank value is an integer representable in $O(n\log n)$ bits. The construction also answers a question of Allman, Lo, and McCormick on polynomial bit length for source-sink monotone two-parameter minimum cuts. Despite this complexity, an optimal contract for quadratic costs always exists with a share of at most $1/2$. Consequently, $O(1/\varepsilon)$ best-response computations suffice for a $(1-\varepsilon)$-approximation. For rational quadratic costs on explicitly described rational polytopes containing zero, this gives a fully polynomial-time approximation scheme. More generally, a share of at most $1/p$ suffices for continuous convex costs homogeneous of degree $p>1$, on any compact convex action set containing zero. For general convex costs with an exact value and bounded-subgradient oracle, we obtain an oracle approximation scheme using approximate best responses and a supplied lower bound on positive welfare.