Walking Through Waypoints

Abstract

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $$G=(V,E)$$G=(V,E), find a shortest walk (“route”) from a source $${s\in V}$$s∈V to a destination $$t\in V$$t∈V that includes all vertices specified by a set $$WP \subseteq V$$WP⊆V: the waypoints. This Waypoint Routing Problem finds immediate applications in the context of modern networked systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable.

On Polynomial Kernels for Traveling Salesperson Problem and its Generalizations

“…The Traveling Salesperson Problem is among the most popular and most intensively studied combinatorial optimization problems in graph theory 1 . From the practical point of view, the problem was already studied in the 1950s; see, e.g., [20].…”

“…A popular practical approach to the single-visit version of TSP is to gradually improve some given solution using local improvements-the so-called q-OPT moves (see, e.g., [44,45]). Marx [54] proved that TSP is W [1]-hard w.r.t. q.…”

“…Moreover, they showed a connection of 4-OPT to APSP. Guo et al [36] considered other suitable measures (parameters) for local search (e.g., r-swap or r-edit) and provided a W [1]-hardness result in general graphs (for both parameters). Furthermore, they gave an FPT algorithm for planar graphs and showed that the existence of polynomial kernels is unlikely even in this case.…”

“…Subset TSP can be solved in time 2 ℓ • n O (1) by first computing the distance between every pair of waypoints, and then solving the resulting ℓ-waypoint instance using the standard Bellman-Held-Karp dynamic programming algorithm [5,40]. Klein and Marx [48] showed that if G is an undirected planar graph with polynomially bounded edge weights, then the problem can be solved significantly faster, in 2 O( √ ℓ log ℓ) • n O (1) time. It is also known that Subset TSP can be solved in 2 O( √ ℓ log ℓ) • n O (1) on edge-weighted directed planar graphs [55].…”

“…Klein and Marx [48] showed that if G is an undirected planar graph with polynomially bounded edge weights, then the problem can be solved significantly faster, in 2 O( √ ℓ log ℓ) • n O (1) time. It is also known that Subset TSP can be solved in 2 O( √ ℓ log ℓ) • n O (1) on edge-weighted directed planar graphs [55]. Amiri et al [1] initiated the study of the variant of WRP discussed in this paper.…”