(Almost) Tight bounds and existence theorems for single-commodity confluent flows
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Bibliographic record
Abstract
A flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge. In this article, we study single-commodity confluent flow problems, where we need to route given node demands to a single destination using a confluent flow. Single- and multi-commodity confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are (multi-commodity) confluent flows since Internet routing is destination based. We present near-tight approximation algorithms, hardness results, and existence theorems for minimizing congestion in single-commodity confluent flows. The maximum edge congestion of a single-commodity confluent flow occurs at one of the incoming edges of the destination. Therefore, finding a minimum-congestion confluent flow is equivalent to the following problem: given a directed graph G with k sinks and non-negative demands on all the nodes of G , determine a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. The main result of this article is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln( k ) in G , if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than H k , the k th harmonic number, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the congestion of an optimal confluent flow to within a factor of (log 2 k )/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand. We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph is k -connected. In particular, we prove that k -connected graphs with k sinks admit confluent flows of congestion less than C + d max , where C is the congestion of the best splittable flow, and d max is the maximum demand of any node in G . The proof of this existence theorem is non-constructive and relies on topological techniques introduced by Lovász.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it