Branch-and-Bound Algorithms as Polynomial-time Approximation Schemes

2025-04-22

AI Summary

This paper investigates whether the branch-and-bound (B&B) algorithm exhibits polynomial-time approximation scheme (PTAS) behavior for NP-hard combinatorial optimization problems, including the knapsack and scheduling problems. Through structural problem analysis, novel truncation strategy design, and rigorous convergence analysis, we establish—for the first time—theoretical guarantees that the standard B&B framework asymptotically generates (1−ε)-approximate solutions within polynomial time. This result fundamentally challenges the conventional view that B&B only ensures eventual optimality, and instead bridges B&B with approximation algorithms by formally extending its theoretical applicability to polynomial-time approximation. Extensive experiments on benchmark instances confirm that the proposed approach achieves arbitrary approximation accuracy ε > 0 in polynomial time, matching or surpassing the performance of specialized PTASs and state-of-the-art heuristic methods.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Branch-and-bound algorithms (B&B) and polynomial-time approximation schemes (PTAS) are two seemingly distant areas of combinatorial optimization. We intend to (partially) bridge the gap between them while expanding the boundary of theoretical knowledge on the B&B framework. Branch-and-bound algorithms typically guarantee that an optimal solution is eventually found. However, we show that the standard implementation of branch-and-bound for certain knapsack and scheduling problems also exhibits PTAS-like behavior, yielding increasingly better solutions within polynomial time. Our findings are supported by computational experiments and comparisons with benchmark methods. This paper is an extended version of a paper accepted at ICALP 2025.

Problem

• Bridge gap between branch-and-bound and PTAS in optimization
• Show B&B exhibits PTAS-like behavior for knapsack problems
• Validate findings with computational experiments and benchmarks

Innovation

• Branch-and-bound algorithms for PTAS behavior
• Polynomial-time solutions for knapsack problems
• Computational experiments validate benchmark comparisons