An Approximation Algorithm for Graph Label Selection

Classifiers can make identical predictions yet require labels to compare their selective performance: confidence ranks weight the same errors differently. We quantify this requirement for the area under the generalized risk-coverage curve (AUGRC). A prelabel lower bound rules out insufficient budgets. With all labels known, a covering linear program bounds the minimum number of labels sufficient to fix the winner (the certificate size) within $K-1$ labels for $K$ candidates. For fixed $K$, independent uniform orders and identical predictions, the prelabel bound approaches one quarter of the pool. With iid Bernoulli errors independent of the orders, every exact acquisition policy reads almost all labels asymptotically, although a two-candidate certificate needs only half. Across 108 feature-panel comparisons on nine datasets, disagreement labels settle every accuracy choice but no AUGRC choice. A 20% budget is ruled out in 96 conditions; certificates need 56-57% on average. On ten conditions with pretrained image classifiers, confidence-score choice reads 68-91% of 10,000 labels for exact selection and 50-67% with AUGRC tolerance $5\times10^{-4}$. An exact stopping test works with any acquisition order. Together, these results link confidence ranks to label budgets and certified model comparison.

The Graph Coloring Problem (GCP) is NP-hard and DSATUR stands as one of the fastest heuristics for it despite producing colorings that typically use more colors than state-of-the-art coloring algorithms. We propose SSLD (Semidefinite Spectral Learning with DSATUR), which improves DSATUR by preprocessing a first good color class before letting DSATUR complete coloring the rest of the given graph. We obtain this color class from a Semidefinite Programming (SDP), similar to an SDP used to compute the Lovász theta number. To the best of our knowledge, SSLD is the first approach to improve DSATUR by preprocessing through fixed color classes. We evaluate SSLD against DSATUR and against a naive 1-color-class preprocessing algorithm on DIMACS instances, random graphs (Erdős--Rényi, Watts-Strogatz, Barabási--Albert), Frequency Assignment and Job Shop Scheduling instances. SSLD matches or beats DSATUR in almost every case across over 1600 benchmark instances, and out performs the naive GISD baseline, allows us to confirm the value brought by the SDP-guided choice of the first color class. This quality comes at a runtime cost of roughly 195 times slower that DSATUR, but demonstrating that SDP-guided preprocessing of a first color class is a direction for future improvements.

This paper studies learning-augmented online weighted vertex cover with advice and a parameter $λ\in (0,1)$. We consider two graph cases: bipartite graphs and general graphs. In both settings, the online algorithm must maintain a feasible vertex cover under irrevocable decisions. We show that these problems admit the same robustness--consistency tradeoffs as learning-augmented ski rental. For the bipartite graph model, we give a randomized algorithm that is $\frac{1}{1-e^{-λ}}$-robust and $\fracλ{1-e^{-λ}}$-consistent. For the general graph model, we give a deterministic algorithm that is $(1+\frac{1}λ)$-robust and $(1+λ)$-consistent. We prove that the tradeoffs above are optimal in both settings. We also validate the proposed algorithms through experiments on synthetic and real-world datasets.