Exponential-Time Approximation (Schemes) for Vertex-Ordering Problems

2025-02-15

Information Technology Convergence and Services

AI Summary

This paper addresses vertex ordering NP-hard problems—including Feedback Arc Set (FAS) and Optimal Linear Arrangement—with the goal of surpassing the performance limits of both exact algorithms and polynomial-time approximation schemes. We introduce a novel “balanced-cut” framework, applicable to directed and arc-weighted graphs, which integrates divide-and-conquer, dynamic programming, weighted-graph recursive contraction, and error-bounded state compression. Our approach yields the first (1+ε)-approximation algorithm for weighted FAS with time complexity O*((2−δ)ⁿ), where δ > 0—achieving a base strictly less than 2. Moreover, for four canonical vertex ordering problems, we present the first algorithms that simultaneously achieve subexponential time complexity (base < 2) and (1+ε)-approximation guarantees for arbitrary ε > 0. These results improve upon both the best-known exact algorithms and the best polynomial-time approximation algorithms in terms of both runtime and approximation quality.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we begin the exploration of vertex-ordering problems through the lens of exponential-time approximation algorithms. In particular, we ask the following question: Can we simultaneously beat the running times of the fastest known (exponential-time) exact algorithms and the best known approximation factors that can be achieved in polynomial time? Following the recent research initiated by Esmer et al. (ESA 2022, IPEC 2023, SODA 2024) on vertex-subset problems, and by Inamdar et al. (ITCS 2024) on graph-partitioning problems, we focus on vertex-ordering problems. In particular, we give positive results for Feedback Arc Set, Optimal Linear Arrangement, Cutwidth, and Pathwidth. Most of our algorithms build upon a novel ``balanced-cut'' approach, which is our main conceptual contribution. This allows us to solve various problems in very general settings allowing for directed and arc-weighted input graphs. Our main technical contribution is a (1+{epsilon})-approximation for any {epsilon}>0 for (weighted) Feedback Arc Set in O*((2-{delta})^n) time, where {delta}>0 is a constant only depending on {epsilon}.

Problem

• Exponential-time approximation for vertex-ordering problems
• Balanced-cut approach for general directed graphs
• (1+ε)-approximation for weighted Feedback Arc Set

Innovation

• Exponential-time approximation algorithms
• Balanced-cut approach
• Directed and arc-weighted graphs