ConRep4CO: Contrastive Representation Learning of Combinatorial Optimization Instances across Types

Considerable efforts have been devoted to machine learning (ML) for combinatorial optimization (CO) problems, especially on graphs. Compared to the active and well-established research for representation learning of text and vision, etc., it remains under-studied for the representation learning of CO problems, especially across different types. In this paper, we try to fill this gap (especially for NP-complete (NPC) problems, as they, in fact, can be reduced to one another). Our so-called ConRep4CO framework, performs contrastive learning by first transforming CO instances in various original forms into the form of Boolean satisfiability (SAT). This scheme is readily doable, especially for NPC problems, including those practical graph decision problems (GDPs) which are inherently related to their NP-hard optimization versions. Specifically, each positive pair of instances for contrasting consists of an instance in its original form and its corresponding transformed SAT form, while the negative samples are other instances not in correspondence. Extensive experiments on seven GDPs (most of which are NPC) show that ConRep4CO significantly improves the representation quality and generalizability to problem scale. Furthermore, we conduct extensive experiments on NP-hard optimization versions of the GDPs, including MVC, MIS, MC and MDS. The results show that introducing ConRep4CO can yield performance improvements of 61.27%, 32.20%, 36.46%, and 45.29% in objective value gaps compared to problem-specific baselines, highlighting the potential of ConRep4CO as a unified pre-training paradigm for CO problems.