network routing engineers

Abstract: We examine whether the integers admit a topology whose connected subsets are precisely the sets of consecutive integers. The main result gives a construction for every integer lattice {\mathbb{Z}}^{n}, n>0; the answer for Z is then obtained as a corollary. We prove that the neighborhoods determined by the parity of the sum of the coordinates form a basis and that topological connectedness agrees exactly with accessibility by finite paths of unit lattice steps. We also present a deductive audit and the correspondence between this argument and a verification in Lean 4 and Mathlib, including the empty set and the independent formal characterization of integer intervals. The construction and mathematical result originate in Problem 5712, proposed by Dan Marcus and solved in 1970 by the Cleveland State University group; the present article contributes a detailed exposition and a report on formal verification.

Abstract: The A* search is a fundamental path-finding algorithm in artificial intelligence. While admissible and consistent heuristics guarantee efficient performance by expanding each state at most once, modern search applications frequently employ powerful but inconsistent heuristics derived from machine learning, randomized evaluations, etc. A long-standing theoretical barrier to using these inconsistent heuristics is the risk of catastrophic node re-expansion, which yields a worst-case exponential time complexity of $Ω(2^n)$. However, empirical observations contradict this pessimistic bound, demonstrating that inconsistent A* operates highly efficiently in practice. To bridge this significant gap between theory and practice, this paper presents the first smoothed analysis of the A* algorithm using inconsistent heuristics. We model typical real-world noise by applying slight random perturbations to the edge weights of worst-case search graphs. Our main result proves that the expected smoothed time complexity of inconsistent A* is bounded by a polynomial, specifically a total iteration number of $O(n^2 m κ)$, where $n$ is the number of nodes, $m$ is the number of edges, and $κ$ controls the scale of random perturbations. Furthermore, we also show that this result naturally extends to the functionally equivalent problem of Dijkstra's algorithm on negative-weight graphs.

Abstract: We study a graph simplification problem introduced by Bernstein, Bodwin, and Wein [ITCS'24]. We start with a graph with arbitrarily large positive edge weights and the goal is to reweight the edges to small aspect ratio (ratio between largest and smallest weight) while preserving the shortest paths structure (the sequence of vertices and edges along shortest paths). They studied whether polynomial aspect ratio is always possible. They proved that for general graphs, both directed and undirected, it is not: there exist graphs for which any shortest-paths preserving reweighting requires exponential aspect ratio. In contrast, they showed that every DAG (directed acyclic graph) admits a reweighting with linear aspect ratio. However, the resulting edge weights are not integers. This motivated them to pose the open question of whether all DAGs admit a reweighting with polynomially-bounded integer edge weights. Our main result is to answer this question in the negative: we prove that there exist DAGs for which any shortest-paths preserving integer reweighting requires weights of size $2^{Ω(n)}$. In fact, this is even true when the DAG has very simple structure: 3 layers of vertices with only 3 vertices in the middle layer. In contrast, we show that if the number of vertices in the middle layer is decreased to 2, then a linear upper bound is possible. We extend our exponential lower bound to the approximate version of the problem where only a single $α$-approximate shortest path in the original graph must be preserved as an exact shortest path in the reweighted graph. Our exponential lower bound holds even for any finite approximation ratio $α>1$.