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Record W2462178547

On approximate min-max theorems for graph connectivity problems

2006· article· en· W2462178547 on OpenAlex
Lap Chi Lau

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueTSpace · 2006
Typearticle
Languageen
FieldComputer Science
TopicCooperative Communication and Network Coding
Canadian institutionsUniversity of Toronto
Fundersnot available
KeywordsCombinatoricsSteiner tree problemMathematicsDisjoint setsVertex (graph theory)Spanning treeConjectureDiscrete mathematicsHypergraphGraph
DOInot available

Abstract

fetched live from OpenAlex

Given an undirected graph G and a subset of vertices S ` V (G), we call the vertices in S the terminal vertices and the vertices in V (G)- S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high "connectivity " among the terminal vertices. The first problem is the Steiner Tree Packing problem, where a Steiner tree is a tree that connects the terminal vertices (Steiner vertices are optional). The goal of this problem is to find a largest collection of edge-disjoint Steiner trees. The second problem is the Steiner Rooted-Orientation problem. In this problem, there is a root vertex r among the terminal vertices. The goal is to find an orientation of all the edges in G so that the Steiner rooted-connectivity is maximized in the resulting directed graph D. Here, the Steiner rooted-connectivity is defined to be the maximum k so that the root vertex has k arc-disjoint paths to each terminal vertex in D. Both problems are generalizations of two classical graph theoretical problems: the edge-disjoint s, t-paths problem and the edge-disjoint spanning trees problem. Polynomial time algorithms and exact min-max relations are known for the classical problems. However, both problems that we study are NP-complete, and thus exact min-max relations are not expected. In the following, we say S is l-edge-connected in G if we need to remove at least l edges in order to disconnect two vertices in S. Clearly, the maximum iii l for which S is l-edge-connected in G is an upper bound on the optimal value for both problems that we study (i.e. the number of edge-disjoint Steiner trees, and the Steiner rooted-connectivity in an orientation). The main result of the Steiner Tree Packing problem is the following approximate min-max relation:

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Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.000
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: none
Teacher disagreement score0.962
Threshold uncertainty score0.402

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0000.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0010.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

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.

Opus teacher head0.035
GPT teacher head0.303
Teacher spread0.268 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it