Adaptive Monte Carlo Search for Conjecture Refutation in Graph Theory

Graph theory is an interdisciplinary field of study that has various applications in mathematical modeling and computer science. Research in graph theory depends on the creation of not only theorems but also conjectures. Conjecture-refuting algorithms attempt to refute conjectures by searching for counterexamples to those conjectures, often by maximizing certain score functions on graphs. This study proposes a novel conjecture-refuting algorithm, referred to as the adaptive Monte Carlo search (AMCS) algorithm, obtained by modifying the Monte Carlo tree search algorithm. Evaluated based on its success in finding counterexamples to several graph theory conjectures, AMCS outperforms existing conjecture-refuting algorithms. The algorithm is further utilized to refute six open conjectures, two of which were chemical graph theory conjectures formulated by Liu et al. in 2021 and four of which were formulated by the AutoGraphiX computer system in 2006. Finally, four of the open conjectures are strongly refuted by generalizing the counterexamples obtained by AMCS to produce a family of counterexamples. It is expected that the algorithm can help researchers test graph-theoretic conjectures more effectively.

An Asymmetric Contrastive Loss for Handling Imbalanced Datasets

Contrastive learning is a representation learning method performed by contrasting a sample to other similar samples so that they are brought closely together, forming clusters in the feature space. The learning process is typically conducted using a two-stage training architecture, and it utilizes the contrastive loss (CL) for its feature learning. Contrastive learning has been shown to be quite successful in handling imbalanced datasets, in which some classes are overrepresented while some others are underrepresented. However, previous studies have not specifically modified CL for imbalanced datasets. In this work, we introduce an asymmetric version of CL, referred to as ACL, in order to directly address the problem of class imbalance. In addition, we propose the asymmetric focal contrastive loss (AFCL) as a further generalization of both ACL and focal contrastive loss (FCL). Results on the FMNIST and ISIC 2018 imbalanced datasets show that AFCL is capable of outperforming CL and FCL in terms of both weighted and unweighted classification accuracies. In the appendix, we provide a full axiomatic treatment on entropy, along with complete proofs.