Polynomial-Time Approximation Schemes for Shortest Path with Alternatives

Abstract

Consider the generic situation that we have to select k alternatives from a given ground set, where each element in the ground set has a random arrival time and cost. Once we have done our selection, we will greedily select the first arriving alternative, and the total cost is the time we had to wait for this alternative plus its random cost. Our motivation to study this problem comes from public transportation, where each element in the ground set might correspond to a bus or train, and the usual user behavior is to greedily select the first option from a given set of alternatives at each stop. We consider the arguably most natural arrival time distributions for such a scenario: exponential distributions, uniform distributions, and distributions with mon. decreasing linear density functions. For exponential distributions, we show how to compute an optimal policy for a complete network, called a shortest path with alternatives, in ${\mathcal O}( n ( \log n + \delta^3 ) )$ time, where n is the number of nodes and δ is the maximal outdegree of any node, making this approach practicable for large networks if δ is relatively small. Moreover, for the latter two distributions, we give PTASs for the case that the distribution supports differ by at most a constant factor and only a constant number of hops are allowed in the network, both reasonable assumptions in practice. These results are obtained by combining methods from low-rank quasi-concave optimization with fractional programming. We finally complement them by showing that general distributions are NP-hard.