Combinatorial Algorithms for Rooted Prize-Collecting Walks and Applications to Orienteering and Minimum-Latency Problems

We consider the rooted prize-collecting walks (PCW) problem, wherein we seek a collection $$\mathcal {C}$$ of rooted walks having minimum prize-collecting cost, which is the (total cost of walks in $$\mathcal {C}$$ ) + (total node-reward of the nodes not visited by any walk in $$\mathcal {C}$$ ). This problem arises naturally as the Lagrangian relaxation of both orienteering (find a length-bounded walk of maximum reward), and the $$\ell $$ -stroll problem (find a minimum-length walk covering at least $$\ell $$ nodes). Our main contribution is to devise a simple, combinatorial algorithm for the PCW problem that returns a rooted tree whose prize-collecting cost is at most the optimum value of the prize-collecting walks problem. This result applies also to directed graphs, and holds for arbitrary nonnegative edge costs. We present two applications of our result. We utilize our algorithm to develop combinatorial approximation algorithms for two fundamental vehicle-routing problems ( $$\mathsf {VRP}$$ s): (1) orienteering; and (2) k-minimum-latency problem ( $$k\text {-}\mathsf {MLP} $$ ), wherein we seek to cover all nodes using k paths starting at a prescribed root node, so as to minimize the sum of the node visiting times. Our combinatorial algorithm allows us to sidestep the part where we solve a preflow-based LP in the LP-rounding algorithms of [13] for orienteering, and in the state-of-the-art 7.183-approximation algorithm for $$k\text {-}\mathsf {MLP} $$ in [17]. Consequently, we obtain combinatorial implementations of these algorithms (with the same approximation factors). Compared to algorithms that achieve the current-best approximation factors for orienteering and $$k\text {-}\mathsf {MLP} $$ , our algorithms have substantially improved running time, and achieve approximation guarantees that match ( $$k\text {-}\mathsf {MLP} $$ ), or are slightly worse (orienteering) than the current-best approximation factors for these problems. We report various computational results for our resulting orienteering algorithms showing that they perform quite well in practice.