The single-source unsplittable flow (SSUF) problem asks to send flow from a common source to terminals with unrelated demands, each terminal being served through a single path. The classical SSUF objective is to minimize the violation of some given arc capacities. A seminal result of Dinitz, Garg, and Goemans showed that, whenever a fractional flow exists respecting the capacities, then there is an unsplittable one violating the capacities by at most the maximum demand. Goemans conjectured a natural cost version of the same result, where the unsplittable flow is required to be no more expensive than the fractional one. Intriguingly, there are arguably no non-trivial graph classes for which it is known to hold. We show that a slight weakening of it holds for planar graphs, by exploiting a connection to a highly structured discrepancy problem. Moreover, our techniques extend to simultaneous upper and lower bounds on the flow values. This affirmatively answers a conjecture of Morell and Skutella for planar SSUF. Finally, we show that our approach can be extended to general (non-planar) graphs with a capacity violation that depends on the genus.
In the Directed Steiner Network (DSN) problem we are given a directed graph and a set of demands $(s_i,t_i)$, and asked to find a cheap subgraph connecting each terminal pair. In its online version, the demands arrive online and must be served by buying edges irrevocably. DSN is a fundamental hard problem in network de...
Gary Hoppenworth, Yao-Wei Long, S. Mahabadi et al.· 0 citations
We give an optimal solution to the Maximum All Request Path Grooming (MARPG) problem motivated by a traffic grooming application and by its interest in computing lower bounds on the cutwidth of a graph. We are given a directed path on vertices and a positive integer capacity (grooming factor). The MARPG problem consi...
J. Bermond, Michel Cosnard, D. Coudert et al.· Networks· 1 citation
We study a graph simplification problem introduced by Bernstein, Bodwin, and Wein [ITCS'24]. We start with a graph with arbitrarily large positive edge weights and the goal is to reweight the edges to small aspect ratio (ratio between largest and smallest weight) while preserving the shortest paths structure (the seque...
It is proved that Minimal-to-Maximal Conversion Search is in fact not output-polynomial and the lower bound construction motivates a more detailed analysis of how certain heuristic choices in the algorithm design affect the running time.
A new proof that the multiple-unicast conjecture holds for networks with at most six coding nodes, without computer-aided search, is given, and it is shown that if the conjecture holds on $\Gamma_{3,3}$, then it holds whenever no three sessions have six distinct terminal locations.
Sirui Liu, Linfeng Que, Zongpeng Li et al.· 0 citations
It is shown that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order, and under the order-dependent model, where bidirectional Dijkstra is not instance-optimal.
Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup et al.· 1 citation
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