Front-to-Attractors: Modifying the Front-to-Front Heuristic in Bidirectional Search

Bidirectional heuristic search can potentially reduce search effort for problems amenable to backward search. Therein, it is well-known that front-to-front heuristics can reduce the number of node expansions, but their overhead is so high that overall runtime almost always increases. We propose BiXDFBnB, a bidirectional depth-first branch-and-bound algorithm that adapts the Single-Frontier Bidirectional Search (SFBDS) framework - originally developed for shortest-path (MIN) problems - to the Generalized Longest Simple Path (GLSP) setting. Because SFBDS inherently operates on paired states, front-to-front (F2F) heuristic evaluation arises naturally and avoids the overhead typically associated with bidirectional frontier management. We show that this adaptation can be successfully applied to maximization (MAX) problems while efficiently handling overlapping constraints. BiXDFBnB is applied to several types of longest-path problems: Longest Simple Path (LSP), Snakes, and Coil-in-the-Box (CIB). Empirical evaluation shows that the new algorithm frequently reduces the number of node expansions and, in some cases, also improves overall runtime.

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.

We focus on the problem of efficient anytime kinodynamic planning for systems with complex dynamics in unstructured environments that make precomputing motion primitives infeasible. Directly applying A* to such problems is computationally infeasible due to the curse of dimensionality. Methods such as Hybrid A* addressed this burden by discretizing the state space, but in turn creating a coupling between tree discovery and the discretization resolution. The Incremental Generalized Hybrid A* (IGHA*) performs search over a hierarchy of resolutions in an anytime fashion to break this coupling, by freezing vertices to use in later search iterations rather than pruning them. However, the frozen vertices can hide solution-supporting vertices from the search at a particular iteration. While classical bidirectional search is motivated by the reduction of search depth, extending IGHA* into the bidirectional setting (termed Bi-IGHA*) obtains additional benefit by fundamentally mitigating the behaviour induced by frozen vertices hiding solutions. We show that Bi-IGHA* preserves IGHA*'s guarantees on monotonic cost improvement and termination. We empirically show that Bi-IGHA* substantially reduces expansions on R3, R4, and R6 planning problems, and achieves equivalent closed-loop performance with kinodynamic planning for high-speed off-road autonomy while requiring significantly fewer expansions. Website: https://personalrobotics.github.io/IGHAStar/biighastar.html