[Paper Review] Front-to-End Bidirectional Heuristic Search with Near-Optimal Node Expansions

Jingwei Chen, Robert C. Holte+2 more|arXiv (Cornell University)|Mar 10, 2017

Metaheuristic Optimization Algorithms Research12 references4 citations

TL;DR

This paper introduces Near-Optimal Bidirectional Search (NBS), a front-to-end bidirectional heuristic search algorithm that guarantees no more than twice the minimum number of node expansions required to cover all 'surely expanded' state pairs under consistent heuristics. NBS achieves near-optimality by modeling necessary expansions as a vertex cover problem and using an efficient approximation algorithm, proving it is optimal in the worst case and outperforming A* and other bidirectional methods on hard or poorly-heuristic problems.

ABSTRACT

It is well-known that any admissible unidirectional heuristic search algorithm must expand all states whose $f$-value is smaller than the optimal solution cost when using a consistent heuristic. Such states are called "surely expanded" (s.e.). A recent study characterized s.e. pairs of states for bidirectional search with consistent heuristics: if a pair of states is s.e. then at least one of the two states must be expanded. This paper derives a lower bound, VC, on the minimum number of expansions required to cover all s.e. pairs, and present a new admissible front-to-end bidirectional heuristic search algorithm, Near-Optimal Bidirectional Search (NBS), that is guaranteed to do no more than 2VC expansions. We further prove that no admissible front-to-end algorithm has a worst case better than 2VC. Experimental results show that NBS competes with or outperforms existing bidirectional search algorithms, and often outperforms A* as well.

Motivation & Objective

• To address the lack of provably optimal front-to-end bidirectional heuristic search algorithms despite known sufficient conditions for node expansion.
• To identify a theoretical lower bound on the number of node expansions required for any admissible front-to-end bidirectional search.
• To design an algorithm that guarantees expansion counts within a factor of two of this lower bound.
• To prove that no admissible front-to-end algorithm can perform better than a 2× factor in the worst case.
• To empirically validate NBS against A*, BS*, MMe, and MM0 across diverse domains, showing consistent competitiveness or superiority.

Proposed method

• The paper derives a graph-theoretic characterization of 'surely expanded' (s.e.) state pairs in bidirectional search, showing that their coverage corresponds to a vertex cover in a derived graph.
• It defines VC as the size of the minimum vertex cover in this graph, establishing it as a lower bound on the number of node expansions required.
• NBS uses a modified version of the Papadimitriou and Steiglitz vertex cover approximation algorithm to select nodes for expansion.
• The algorithm performs front-to-end bidirectional search, expanding nodes only when they are part of the computed vertex cover, ensuring coverage of all s.e. pairs.
• It maintains admissibility by using consistent heuristics and ensures no redundant expansions through careful frontier management.
• Theoretical analysis proves that NBS never expands more than 2×VC nodes, and that no admissible front-to-end algorithm can guarantee better than a 2× worst-case bound.

Experimental results

Research questions

• RQ1 What is the theoretical minimum number of node expansions required to cover all 'surely expanded' state pairs in front-to-end bidirectional heuristic search?
• RQ2 Can a front-to-end bidirectional search algorithm be designed to achieve near-optimality in terms of node expansions, with a provable worst-case bound of 2× the theoretical minimum?
• RQ3 Is there a fundamental worst-case performance limit for any admissible front-to-end bidirectional search algorithm, and if so, what is it?
• RQ4 How does the performance of a near-optimal algorithm like NBS compare to A* and other bidirectional algorithms on problems with weak heuristics or high difficulty?
• RQ5 To what extent do expansions with f = C* (optimal cost) affect the performance gap between NBS and A* in practice?

Key findings

• NBS guarantees to expand no more than twice the size of the minimum vertex cover (VC), which is the theoretical lower bound on necessary expansions.
• No admissible front-to-end bidirectional algorithm can have a worst-case expansion bound better than 2×VC, making NBS optimal in the worst-case sense.
• On hard problems and with weak heuristics—especially in Tower of Hanoi and pancake puzzles—NBS significantly outperforms A* and other bidirectional algorithms.
• In grid-based pathfinding (e.g., brc203d maps), NBS performs competitively with A* and BS*, though it is about 30% slower due to overhead.
• On problems with strong heuristics like GAP in pancake puzzles, NBS expands many nodes with f = C*, which explains its higher expansion count despite theoretical bounds.
• In 30 instances from the brc203d domain, NBS performs more than twice as well as other algorithms in terms of expansion count, demonstrating practical superiority beyond the theoretical worst-case bound.