Algorithms for Orienteering and DiscountedReward TSP Shuchi Chawla

Algorithms for Orienteering and Discounted-Reward TSP Shuchi Chawla Carnegie Mellon University Joint work with Avrim Blum, Adam Meyerson, David Karger, Minkoff and Terran Lane Shuchi Chawla, Carnegie Mellon University Maria

The focus of our paper w Given weighted graph G, root s, reward on nodes v w Construct a path P rooted at s w High level objective: Collect large reward in little time n Orienteering Maximize reward collected with path of length D n Discounted-Reward TSP Reward from node v, if reached at time t is v t reward Orienteering Dis. Rew. TSP 2 time Shuchi Chawla, Carnegie Mellon University

The focus of our paper w Given weighted graph G, root s, reward on nodes v w Construct a path P rooted at s w No approximation algorithm known previously for the rooted non-geometric versionreward in little time High level objective: Collect large n Orienteering Maximize reward collected with path of length D n Discounted-Reward TSP New problem Reward from node v, if reached at time t is v t A related problem… n K-Traveling Salesperson [Garg] [Arora. Karpinski] … Best: (2+ )-approx Minimize length while collecting at least K in reward 3 Shuchi Chawla, Carnegie Mellon University

Our contributions Problem K-path ( CP) Min-Excess Path ( EP) Orienteering Discounted-Reward TSP 4 Source/Reduction Approximation [Chaudhuri et al’ 03] 2+ 1. 5 CP – 0. 5 2. 5+ 1+[ EP] 4 (1+ EP)(1+1/ EP) EP 2+ 8. 12+ 6. 75+ Shuchi Chawla, Carnegie Mellon University

A Robot Navigation Problem w Task: deliver packages to locations in a building w Faster delivery => greater happiness w Classic formulation – Traveling Salesperson Problem Find the shortest tour covering all locations w Uncertainty in robot’s lifetime/behavior n n battery failure; sensor error… Robot may fail before delivering all packages w Deliver as many packages as possible n 5 Some packages have higher priority than others Shuchi Chawla, Carnegie Mellon University

Robot Navigation: A probabilistic view w At every time step, the robot has a fixed probability (1 - ) of failing w If a package with value is delivered at time t, the expected reward is t “Discounted Reward” w Goal: Construct a path such that the total discounted reward collected is maximized Discounted-Reward TSP Alternately, robot has a fixed battery life D Goal: Construct path of length at most D that collects maximum reward Orienteering 6 Shuchi Chawla, Carnegie Mellon University

Rest of this talk w The Min-Excess problem w Using Min-Excess to solve Orienteering w Solving Min-Excess w Using Min-Excess to solve Discounted-Reward TSP w Extensions and open problems 7 Shuchi Chawla, Carnegie Mellon University

Using K-path directly w First attempt – Use distance-based approximations to approximate reward w Let OPT(d) = max achievable reward with length d w A 2 -approx for distance implies that ALG(d) ¸ OPT(d/2) w However, we may have OPT(d/2) << OPT(d) w Bad trade-off between distance and reward! w Same problem with Discounted-Reward TSP 8 Shuchi Chawla, Carnegie Mellon University

Approximating Orienteering w Using a distance-based approx n n Divide the optimal path into many segments Approximate the max reward segment using distance saved by short-cutting other segments w If min-distance between s and v is d, we spend at least d in going to v, regardless of the path s 9 t Shuchi Chawla, Carnegie Mellon University

Approximating Orienteering w Using a distance-based approx n n Divide the optimal path into many segments Approximate the max reward segment using distance saved by short-cutting other segments w If min-distance between s and v is d, we spend at least d in going to v, regardless of the path w Approximate the “extra” length taken by a path over the shortest path length Min-Excess Path Problem w If OPT obtains k reward with length d+ , ALG should obtain the same reward with length d+ 10 Shuchi Chawla, Carnegie Mellon University

From Min-Excess to Orienteering w There exists a path from s to t, that n n n collects reward at least has length · D t is the farthest from s among all nodes in the path w Excess at node v = “ v” = extra time taken to reach v = d. Pv – dv 2 t/3 t s new excess = t/3 11 t· D-dt Can afford an excess up to t Shuchi Chawla, Carnegie Mellon University

From Min-Excess to Orienteering w There exists a path from s to t, that n n n collects reward at least has length · D t is the farthest from s among all nodes in the path Using an r-approx for Min-excess, we get an rw For any integer r, 9 a path from s to v that approximation n collects reward /r Note: If t is not the farthest node, a similar n has excess · (D-dt)/r · (D-dv)/r analysis gives an r+1 approximation 2 t/3 t s new excess = t/3 12 t· D-dt Can afford an excess up to t Shuchi Chawla, Carnegie Mellon University

Solving Min-Excess w OPT = d+ ; k-path gives us ALG = (d+ ) We want ALG = d + w Note: When ¼ d, (d+ ) ¼ d + O( ) w Idea: When is large, approximate using k-path w What if << d ? w Small path is almost like a shortest path or “its distance from s mostly increases monotonically” 13 Shuchi Chawla, Carnegie Mellon University

Solving Min-Excess w OPT = d+ ; k-path gives us ALG = (d+ ) We want ALG = d + w Note: When ¼ d, (d+ ) ¼ d + O( ) w Idea: When is large, approximate using k-path segments using dynamic programming w What if << d Patch ? w Small path is almost like a shortest. Approximate path or “its distance from s mostly increases monotonically” t s Dynamic Program w Idea: Completely monotone path use dynamic wiggly programming! monotone 14 Shuchi Chawla, Carnegie Mellon University

Solving Discounted-Reward TSP w WLOG, = ½. Reward of v at time t = v t w An interesting observation: OPT collects half of its reward before the first half life node that has excess 1 w Therefore, approximate the min-excess from s to v w New path has excess 3. Reward by factor of 23. 16 -approximation ’ = 2 OPT(v, t) > OPT reward ¸ OPT/2 s v t length of entire remaining path decreases by 1 excess = 1 15 Shuchi Chawla, Carnegie Mellon University

A summary of our results Problem K-path ( CP) Min-Excess Path ( EP) Orienteering Discounted-Reward TSP 16 Source/Reduction Approximation [Chaudhuri et al’ 03] 2+ 1. 5 CP – 0. 5 2+ 1+[ EP] 4 (1+ EP)(1+1/ EP) EP 6. 75+ Shuchi Chawla, Carnegie Mellon University

Some extensions w Unrooted versions w Multiple robots w Max-reward Steiner tree of bounded size 17 Shuchi Chawla, Carnegie Mellon University

Future work… w Improve the approximations n 2 -approx for Orienteering? w Robot Navigation n A highly complex process with various kinds of uncertainty n Can we model the MDP as a simple graph problem? w Different deadlines for different packages 18 Shuchi Chawla, Carnegie Mellon University