Approximation Algorithms for Orienteering and Discounted-Reward TSP Blum, Chawla, Karger, Lane, Meyerson, Minkoff CS 599: Sequential Decision Making in Robotics University of Southern California Spring 2011

TSP: Traveling Salesperson Problem Graph V, E Find a tour (path) of shortest length that visits each vertex in V exactly once Corresponding decision problem – Given a tour of length L decide whether a tour of length less than L exists – NP-complete Highly likely that the worst case running time of any algorithm for TSP will be exponential in |V|

Robot Navigation Can’t go everywhere, limits on resources Many practical tasks don’t require completeness but do require immediacy or at least some notion of timeliness/urgency (e.g. some vertices are short-lived and need to get to them quickly)

Prizes, Quotas and Penalties Prize Collecting Traveling Salesperson Problem (PCTSP) – A known prize (reward) available at each vertex – Quota: The total prize to be collected on the tour (given) – Not visiting a vertex incurs a known penalty – Minimize the total travel distance plus the total penalty, while starting from a given vertex and collecting the pre-specified quota – Best algorithm is a 2 approximation Quota TSP – All penalties are set to zero – Special case is k-TSP, in which all prizes are 1 (k is the quota) – k-TSP is strongly tied to the problem of finding a tree of minimum cost spanning any k vertices in a graph, called the k-MST problem Penalty TSP: no required quota, only penalties All these admit a budget version where a budget is given as input and the goal is to find the largest k-TSP (or other) whose cost is no more than the budget

Orienteering Orienteering: Tour with maximum possible reward whose length is less than a pre-specified budget B o  ri  en  teer  ing |ˌôriənˈti(ə)ri NG |noun a competitive sport in which participants find their way to various checkpoints across rough country with the aid of a map and compass, the winner being the one with the lowest elapsed time. ORIGIN 1940s: from Swedish orientering.

Approximating Orienteering Any algorithm for PC-TSP extends to unrooted Orienteering Thus best solution for unrooted Orienteering is at worst a 2 approximation No previous algorithm for constant factor approximation of rooted Orienteering

Discounted-Reward TSP Undirected weighted graph Edge weights represent transit time over the edge Prize (reward) on vertex v Find a path visiting each vertex at time that maximizes

Discounting and MDPs Encourages early reward collection, important if conditions might change suddenly Optimal strategy is a policy (a mapping from states to action) Markov decision process – Goal is to maximize the expected total discounted reward (can be solved in polynomial time) in a stochastic action setting – Can visit states multiple times Discounted-Reward TSP – Visit a state only once (reward available only on first visit) – Deterministic actions

Overall Strategy Approximate the optimum difference between the length of a prize-collecting path and the length of the shortest path between its endpoints Paper gives – An algorithm that provably gives a constant factor approximation for this difference – A formula for the approximation The results mean that constant factor approximations exist (and can be computed) for Orienteering and Discounted-Reward TSP

Path Excess Excess of a path P from s to t: Minimum excess path of total prize is also the minimum cost path of total prize An (s,t) path approximating optimal excess by factor will have length (by definition) Thus a path that approximates min excess by will also approximate minimum cost path by

Results ProblemApproximation factorSource k-TSPKnown from prior work (best value is 2) Min-excessThis paper OrienteeringThis paper Discounted-Reward TSP (roughly)This paper First letter is objective (cost, prize, excess, or discounted prize) and second is the structure (path, cycle, or tree)

Min Excess Algorithm Let P* be shortest path from s to t with Let Min-excess algorithm returns a path P of length with where

Orienteering Algorithm Compute maximum-prize path of length at most D starting at vertex s 1.Perform a binary search over (prize) values k 2.For each vertex v, compute min-excess path from s to v collecting prize k 3.Find the maximum k such that there exists a v where the min-excess path returned has length at most D; return this value of k (the prize) and the corresponding path

Discounted-Reward TSP Algorithm 1.Re-scale all edge length so 2.Replace each prize by the prize discounted by the shortest path to that node 3.Call this modified graph G’ 4.Guess t – the last node on optimal path P* with excess less than 5.Guess k – the value of 6.Apply min-excess approximation algorithm to find a path P collecting scaled prize k with small excess 7.Return this path as solution

Results ProblemApproximation factorSource k-TSPKnown from prior work (best value is 2) Min-excessThis paper OrienteeringThis paper Discounted-Reward TSP (roughly)This paper First letter is objective (cost, prize, excess, or discounted prize) and second is the structure (path, cycle, or tree)