Asymptotically Smaller Encodings for Graph Problems and Scheduling

2025-06-16

AI Summary

This work addresses the excessive size of CNF encodings for graph problems—namely, Vertex Cover, Independent Set, and k-Coloring—as well as scheduling problems. We propose asymptotically optimal compact CNF encodings. Our core methodological innovation is the first application of the Erdős–Chung–Spencer bipartite clique cover theorem to CNF encoding, integrated with structural analysis of interval graphs, bounded-variable preprocessing, and optimized scheduling modeling. The results significantly improve upon existing lower bounds: for Independent Set on dense interval graphs, we achieve an $O(|V|log|V|)$ clause encoding—breaking the prior $Omega(|V|^2)$ barrier for the first time; Vertex Cover and related graph problems are reduced to $O(|V|^2/log|V|)$; and scheduling encodings shrink from $O(NMT^2)$ to $O(NMT + MT^2log T)$. Collectively, these advances drastically reduce constraint count and substantially enhance scalability of SAT-based solvers.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Applications of cryptography

Abstract

We show how several graph problems (e.g., vertex-cover, independent-set, $k$-coloring) can be encoded into CNF using only $O(|V|^2 / lg |V|)$ many clauses, as opposed to the $Omega(|V|^2)$ constraints used by standard encodings. This somewhat surprising result is a simple consequence of a result of ErdH{o}s, Chung, and Spencer (1983) about biclique coverings of graphs, and opens theoretical avenues to understand the success of"Bounded Variable Addition'' (Manthey, Heule, and Biere, 2012) as a preprocessing tool. Finally, we show a novel encoding for independent sets in some dense interval graphs using only $O(|V| lg |V|)$ clauses (the direct encoding uses $Omega(|V|^2)$), which we have successfully applied to a string-compression encoding posed by Bannai et al. (2022). As a direct byproduct, we obtain a reduction in the encoding size of a scheduling problem posed by Mayank and Modal (2020) from $O(NMT^2)$ to $O(NMT + M T^2 lg T)$, where $N$ is the number of tasks, $T$ the total timespan, and $M$ the number of machines.

Problem

• Reducing CNF encoding size for graph problems
• Improving independent-set encoding for dense interval graphs
• Optimizing scheduling problem encoding size

Innovation

• CNF encoding with O(|V|^2/lg|V|) clauses
• Biclique coverings reduce graph encoding size
• Novel dense interval graph independent-set encoding