Lower Bounds

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Probabilistic Focal Search: Accelerating Bounded-Suboptimal Search via Lower-Bound Advancement
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Sound Probabilistic Safety Bounds for Large Language Models
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Sharp First-Order Lower Bounds for Higher-Order Smooth Nonconvex Optimization
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Auditing Near-Optimal Policies Can Be Exponentially Hard: Conditional Query Lower Bounds via Occupancy Rashomon Capacity
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Lower Bounds and Proximally Anchored SGD for Non-Convex Minimization Under Unbounded Variance
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On the Condition Number Dependency in Bilevel Optimization
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Configuration-Dependent Lower Bounds for Approximation by Shallow ReLU $^k$ Networks on the SphereConfiguration-Dependent Lower Bounds for Approximation by Shallow ReLU^k Networks on the Sphere
Oct 5, 2025Tong Mao, Jinchao XuNeural-Network Approximation Approximation

Sep 12, 2026 · cs.AIJ/K move · Enter open · S save

Probabilistic Focal Search: Accelerating Bounded-Suboptimal Search via Lower-Bound Advancement

Minh Vu Duc, Trung Le Huu, Hà Minh Hoàng, Trung Thanh Nguyen+2

Bounded-suboptimal search seeks a solution within a factor

w

of optimal while reducing search effort. Focal Search (FS) uses heuristic guidance within FOCAL, the frontier nodes eligible under the threshold

w f_{\min}

, but its deterministic policy may leave

f_{\min}

unchanged for many expansions. We introduce Probabilistic Focal Search (PFS), which follows the FS guided choice with probability

p

and expands a minimum-

f

OPEN node with probability

1-p

. The latter branch encourages the lower bound to advance, enlarging FOCAL and admitting nodes that may lead to feasible solutions. By balancing guidance and lower-bound advancement, this mechanism can reduce time to a bounded solution when progress is limited by delayed FOCAL admission. As a secondary transfer experiment, we apply the same scheduler to Dynamic Potential Search, yielding Probabilistic Dynamic Potential Search (PDPS). We benchmark PFS against FS on N-Puzzle, Pancake Sorting, and the Traveling Salesperson Problem (TSP), and evaluate its anytime extension on the Generalized Covering TSP (GCTSP), using multiple

w

and

p

values. Across these benchmarks, the largest gains occur when long

f_{\min}

plateaus delay useful FOCAL admissions; in such settings, the probabilistic factor may reduce node expansions by about 90% or more (e.g., on N-Puzzle and TSP). For the anytime algorithm family, Anytime Probabilistic Focal Search (APFS) outperforms all tested algorithms in evaluating anytime methods on GCTSP. We also observe that the benefit is smaller when the deterministic search already advances efficiently (e.g., Pancake Sorting), indicating that the probabilistic factor is most useful when FOCAL admission is a search bottleneck. The PDPS transfer shows that the mechanism also transfers to potential guidance, although its common-success effects remain domain- and bound-dependent.

Search Algorithms Salesman Problem Lower Bounds

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