Finding Optimal Solutions with Neighborly Help
Bibliographic record
Abstract
Abstract Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighbor instances, that is, instances with one local modification? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems, most notably, graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, colorability and vertex cover. For example, we show that it is $$\text {NP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>NP</mml:mtext> </mml:math> -hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in $$\text {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>P</mml:mtext> </mml:math> . We observe that vertex cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal -3- UnColorability is complete for $$\text {DP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>DP</mml:mtext> </mml:math> (differences of $$\text {NP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>NP</mml:mtext> </mml:math> sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For vertex cover, we show that recognizing $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -vertex-critical graphs is complete for $$\Theta _2^\text {p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Θ</mml:mi> <mml:mn>2</mml:mn> <mml:mtext>p</mml:mtext> </mml:msubsup> </mml:math> (parallel access to $$\text {NP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>NP</mml:mtext> </mml:math> ), obtaining the first completeness result for a criticality problem for this class.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".