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Algorithms for Orienteering and Discounted-Reward TSP

Algorithms for Orienteering and Discounted-Reward TSP

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Algorithms for Orienteering and Discounted-Reward TSP

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

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

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

  4. Our contributions Orienteering Discounted-Reward TSP Problem Source/Reduction Approximation K-path (CP) [Chaudhuri et al’03] 2+ Min-Excess Path (EP) 1.5 CP – 0.5 2.5+ 2+ 1+[EP] 4 (1+EP)(1+1/EP)EP 8.12+ 6.75+ Shuchi Chawla, Carnegie Mellon University

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

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

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

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

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

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

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

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

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

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

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

  16. A summary of our results Orienteering Discounted-Reward TSP Problem Source/Reduction Approximation K-path (CP) [Chaudhuri et al’03] 2+ Min-Excess Path (EP) 1.5 CP – 0.5 2+ 1+[EP] 4 (1+EP)(1+1/EP)EP 6.75+ Shuchi Chawla, Carnegie Mellon University

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

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