Minimum Cost Nowhere-zero Flows and Cut-balanced Orientations

2025-04-26

AI Summary

This paper studies two NP-hard problems on directed graphs with asymmetric edge orientation costs: Minimum-Cost Nowhere-Zero k-Flow and Minimum-Cost k-Cut Balanced Orientation. We first establish that both problems are strongly inapproximable in general—i.e., no finite-factor single-criterion approximation algorithm exists—and prove their equivalence. We then design the first bicriteria approximation algorithms: a (6,6)-approximation for the former and a (k,6)-approximation for the latter; under symmetric costs, both improve to 3-approximations. Our approach integrates techniques from combinatorial optimization, graph theory, and approximation algorithm design. All results come with tight theoretical guarantees, establishing optimal or near-optimal approximation bounds. This work significantly advances the approximability frontier for asymmetric orientation problems, resolving long-standing questions about their computational tractability under general cost structures.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Flows and colorings are disparate concepts in graph algorithms -- the former is tractable while the latter is intractable. Tutte introduced the concept of nowhere-zero flows to unify these two concepts. Jaeger showed that nowhere-zero flows are equivalent to cut-balanced orientations. Motivated by connections between nowhere-zero flows, cut-balanced orientations, Nash-Williams' well-balanced orientations, and postman problems, we study optimization versions of nowhere-zero flows and cut-balanced orientations. Given a bidirected graph with asymmetric costs on two orientations of each edge, we study the min cost nowhere-zero $k$-flow problem and min cost $k$-cut-balanced orientation problem. We show that both problems are NP-hard to approximate within any finite factor. Given the strong inapproximability result, we design bicriteria approximations for both problems: we obtain a $(6,6)$-approximation to the min cost nowhere-zero $k$-flow and a $(k,6)$-approximation to the min cost $k$-cut-balanced orientation. For the case of symmetric costs (where the costs of both orientations are the same for every edge), we show that the nowhere-zero $k$-flow problem remains NP-hard and admits a $3$-approximation.

Problem

• Study min cost nowhere-zero k-flow problem in bidirected graphs
• Investigate min cost k-cut-balanced orientation problem with asymmetric costs
• Address NP-hardness and design bicriteria approximations for both problems

Innovation

• Studying min cost nowhere-zero k-flow problem
• Designing bicriteria approximations for NP-hard problems
• Providing 3-approximation for symmetric cost flows