Skyline-First Traversal as a Control Mechanism for Multi-Criteria Graph Search

We formalize the Steiner Traveling Salesman Problem (Steiner-TSP) on Graphs of Convex Sets (GCS), which seeks a minimum-cost closed trajectory through required convex sets while allowing optional transit vertices and revisits. To explore the resulting infinite solution space, we propose a unified branch-and-bound search over rooted walk prefixes. Additive lower-bound-graph costs bound committed prefixes, while a cut-separated connected-flow relaxation lower-bounds the residual cost of visiting every remaining target and returning to the root. Under a uniform positive-cost assumption, best-first traversal terminates after finitely many expansions on every feasible instance without an initial incumbent, whereas depth-first traversal does so once a finite incumbent is available. For a user-specified factor $ε\geq1$, a global lower bound certifies that either strategy's incumbent cost is at most $ε$ times the global optimum. We further demonstrate joint sensing-mode, visitation-order, and continuous-trajectory selection for a mobile-manipulator inspection task, including action precedences expressed in linear temporal logic over finite traces (LTL$_f$). Both traversal strategies find feasible solutions on all benchmark instances within 30s with mean certified optimality gaps of 28.1% and 29.7%, respectively, whereas two recent baselines succeed on only about half of the instances

In the A* search algorithm, the tie-breaking strategies for nodes with the same $f$-value determines which states A* expands on the final $f$-layer. For nine standard tie-breaking strategies, we show that under a consistent heuristic, every pair has positive-cost instances favoring each strategy over the other by an arbitrarily large additive expansion gap. A parameterized unit-cost grid example also gives unbounded expansion-count ratios between low-$h$ with FIFO and LIFO. In unit-cost search with $h > 0$ at non-goals, exact heuristic values near the goal lead to complementary extremal results: low-$h$ minimizes the number of remaining expansions from a common configuration within the perfect region, while high-$h$ maximizes the total number of expansions when every final-plateau state with $h=1$ is a goal predecessor. Finally, with the evaluation function $f_α = g + αh$, when $h>0$ at non-goals, every heuristic weight $0 \leq α<1$ eliminates tie-breaking sensitivity, and all tie-breaking strategies expand the same set of states.

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.