1. Algorithms for Path-Planning Shuchi Chawla Carnegie Mellon University Jan 17, 2004

2. Algorithms for Path-Planning Shuchi Chawla 2 General research interests Path-planning algorithms Privacy in public databases Clustering with qualitative information Algorithmic game theory Graph partitioning & Sparsest cut

3. Algorithms for Path-Planning Shuchi Chawla 3 A trick-o-treat problem It’s Halloween… Collect as much candy as possible between 6pm and 8pm Goal: Get as much candy as possible In what order should you visit houses?

4. Algorithms for Path-Planning Shuchi Chawla 4 Path-planning Informally… planning and ordering of tasks Classic instance ― Traveling Salesman Problem Find the shortest tour covering all given locations A natural extension ― Orienteering Cover as many locations as possible by a given deadline Many variants, applications:

5. Algorithms for Path-Planning Shuchi Chawla 5 Informally… planning and ordering of tasks Classic instance ― Traveling Salesman Problem A natural extension ― Orienteering Many variants, applications: –Delivery & distribution problems –Production planning, assembly analysis –Robot navigation Solved at large scales in practice Up the elevator Up the stairs Path-planning 1:002:003:004:0012:0011:0010:009:00 Package A, San Jose Package C, Cupertino, *priority* Package B, Mtn. View Office; Pick packages C & DOffice; Pick packages A & B Package D, Palo Alto Pick up and delivery schedule, Monday, Jan 17

6. Algorithms for Path-Planning Shuchi Chawla 6 Path-planning: A brief history Studied in OR for the past 20-30 years (also known as Vehicle Routing) Many other variants: multiple vehicles, stochastic demands, pickup-and-delivery Most are NP-hard A number of heuristics: cutting plane methods  branch and bound simulated annealing  genetic algorithms … Stochastic planning problems in Robotics and AI Poly-time solvable problems e.g. stochastic shortest-paths Linear and Dynamic programming techniques Approximation algorithms: very little known Find a solution with value  (1/  ).OPT in poly-time

7. Algorithms for Path-Planning Shuchi Chawla 7 Approximation Results A reward vs. time trade-off A budget on time; maximize reward –Orienteering –Deadline-TSP –TSP with Time-Windows A quota on reward; minimize time –TSP –k-TSP –Min-Excess Optimize a combination of reward and time –Prize-Collecting TSP –Discounted-Reward TSP single deadline on time different deadlines on different locations different time windows for diff. locations visit all nodes visit k nodes visit k nodes, but minimize excess Minimize time plus reward foregone Max. reward, reward decreases with time

8. Algorithms for Path-Planning Shuchi Chawla 8 A reward vs. time trade-off A budget on time; maximize reward –Orienteering –Deadline-TSP –TSP with Time-Windows A quota on reward; minimize time –TSP –k-TSP –Min-Excess Optimize a combination of reward and time –Prize-Collecting TSP –Discounted-Reward TSP 1.5 [Christofides ’76] 2+  [BRV99] [Garg99] [AK00] [CGRT03] … ? 1.5 [Christofides ’76] 2+  [BRV99] [Garg99] [AK00] [CGRT03] … 2+  Approximation Results ?????? 2 [Goemans Williamson ’92] ? 3 3 log n 3 log 2 n 2 [Goemans Williamson ’92] 6.75+  FOCS’03: Blum, Karger, C., Lane, Meyerson, Minkoff STOC’04: Bansal, Blum, C., Meyerson

9. Algorithms for Path-Planning Shuchi Chawla 9 A road-map An approximation for Orienteering Current and future work –Robot navigation & path-planning in a stochastic world Other research

10. Algorithms for Path-Planning Shuchi Chawla 10 Back to candy-collection (Orienteering) The givens A “map” G of locations, distances “Rewards”  on locations A deadline D To find A path that collects as much reward as possible by deadline D NP-hard; we want to find an approximation Find a solution with reward  (1/  ).OPT in poly-time

11. Algorithms for Path-Planning Shuchi Chawla 11 A greedy approach Visit the closest location; continue until time over Assume for simplicity – all locations have equal reward Local decisions are bad – consider the “big picture” –treat clusters of locations as single entities with large reward Optimal Greedy

12. Algorithms for Path-Planning Shuchi Chawla 12 A greedy approach Back-tracking is not feasible ― the first step could already be disastrous! Suppose we had twice or thrice the time … Several algorithms for k-TSP obtain as much reward as OPT in this case! [BRV99] [Garg99] [AK00] [CGRT03] … Greedy Optimal Algorithm using twice the optimal length

13. Algorithms for Path-Planning Shuchi Chawla 13 k-TSP: Visit reward at least k as fast as possible Suppose OPT has reward = k and length = D Find path of reward = k and length  .D “dual” to Orienteering –Orienteering: length  D; maximize reward –k-TSP: reward  k; minimize length Equivalent at optimality; very different in approximation: –Several approximations for k-TSP : the best ― a (2+  )- approx [Chaudhuri et al. ’03] –No approximations known for Orienteering until recently! k-TSP: Approximating the length An  -approximation to k-TSP

14. Algorithms for Path-Planning Shuchi Chawla 14 Suppose OPT(x) = optimal reward with length x ALG(x) = k-TSP algo’s reward with length x 2-approx to k-TSP  ALG(2x)  OPT(x) Want ALG(D)  (1/  ).OPT(D) However, we may have OPT(D/2) << OPT(D) Bad trade-off between distance and reward! Delving deeper does not help – –algorithms for k-TSP use a “primal-dual” subroutine [Goemans Williamson’92] –inherently aim for the low-hanging fruit first Using k-TSP for Orienteering OPT(D/2) OPT(D) ALG(D) 2-approx to k-TSP  ALG(D)  OPT(D/2)

15. Algorithms for Path-Planning Shuchi Chawla 15 Identify a “rich” portion of OPT and apply your favorite approximation algorithm there Save length from other parts of OPT –Length of OPT = a + b –Length of ALG = dist( home, v) + 2b If dist( home, v)  a, then ALG may have length  D Approximate the “excess” length of OPT ― the amount saved by short-cutting The key insight OPT(D) Reward rich portion of OPT OPT ALG Reward-rich segment v Length up to v Length after v Using a 2-approx for k-TSP Instead, we want: ALG = dist( home, v) + b + extra length saved earlier

16. Algorithms for Path-Planning Shuchi Chawla 16 The Min-Excess problem Visit reward at least k between start s and end t, minimize  (P) = ℓ (P) – dist(s,t) Equivalent to k-TSP at optimality However, approximation is different: –length of OPT =  + dist(s,t) –length of min-excess approx =   + dist(s,t) –length of k-TSP approx=   +  dist(s,t) –Min-excess is strictly harder to approximate than k-TSP A special case: when dist(s,t)  , – length of k-TSP approx    + (  -1)  + dist(s,t) – gives a (2  -1)-approx to min-excess Idea: when  is large, approximate min-excess using k-TSP What if  is small?

17. Algorithms for Path-Planning Shuchi Chawla 17 To find best path for (v,k), try all paths that end at some u and collect reward k-1 Pick the one with smallest excess Approximating Min-Excess The small  case The path is almost like a shortest path –Visits locations roughly in order of increasing distance from home Suppose OPT is monotone, we can find it exactly! Idea #2: When OPT is monotone, use dynamic programming (Note: We still don’t know which ones to visit!) For all locations v, and reward k: find & store best path from home to v with reward k u v u v k=4 k=5

18. Algorithms for Path-Planning Shuchi Chawla 18 Approximating Min-Excess Idea #1: When  is large, approximate min-excess using k-TSP Idea #2: When OPT is monotone, use dynamic programming Gives a (2+  )-approximation for Min-Excess [FOCS’03] Returns path with length (2+  )  + dist(s,t) Use k-TSP Dynamic Program monotone wiggly Patch segments using dynamic programming OPT

19. Algorithms for Path-Planning Shuchi Chawla 19 Construct a path from s to t, that has length  D and collects maximum possible reward Given a 3-approximation to min-excess: 1. Divide into 3 “equal-reward” parts (hypothetically) –Length of OPT = ℓ white +  1 +  2 +  3  D –We can also afford excess up to D - ℓ white   1 +  2 +  3 An algorithm for Orienteering Excess of one of the segments = min {  1,  2,  3 }  (  1 +  2 +  3 )/3 11 22 33 Excess of path from u to v  (u,v) = ℓ (u,v)–d(u,v) An r-approximation for min-excess gives a  r  -approximation for Orienteering v1v1 v2v2 OPT ALG Open: Given an r-approx for min-excess, can we get r-approx to Orienteering? 2. Approximate the part with the smallest excess  3-approximation to orienteering

20. Algorithms for Path-Planning Shuchi Chawla 20 Open problems Better approximations –Vehicle capacities –Precedence constraints Faster approximations How hard are they? –Hardness of approximation results for path-planning –Only small-constant-factor hardness known Stochastic models –Map changes over time –Requests arrive stochastically –Robot Navigation

21. Algorithms for Path-Planning Shuchi Chawla 21 An application: Robot Navigation Robot’s task: deliver packages to various locations Planning with uncertainty –may run out of battery power –may crash into a wall –may face unforeseen obstacle that causes delay Typically modeled as a Markov Decision Process –Current position of the robot summarized as a “state” –Several actions available in each state –Each action results in a new state with some probability

22. Algorithms for Path-Planning Shuchi Chawla 22 An application: Robot Navigation Typically modeled as a Markov Decision Process –Current position of the robot summarized as a “state” –Several actions available in each state –Each action results in a new state with some probability 0.2 0.5 0.3 0.1 0.6 0.3 0.5 0.2 0.8

23. Algorithms for Path-Planning Shuchi Chawla 23 Stochastic path-planning Measure reward/time in expectation Stochastic-TSP: find a strategy that takes minimum possible time in expectation to visit all nodes Harder than the deterministic case: –Some problems are possibly PSPACE-hard –Generalize deterministic “directed” problems My focus –Currently examining approximation for stochastic-TSP joint work with Blum, Kleinberg and McMahan –Generic technique of transforming deterministic-case algorithms to their stochastic versions

24. Algorithms for Path-Planning Shuchi Chawla 24 To summarize… Path-planning: wide variety of highly applicable problems Little known in terms of approximations Provided the first approximations for some of them Important future directions –Closing the gap between hardness and approximation –Stochastic models

25. Algorithms for Path-Planning Shuchi Chawla 25 General research interests Path-planning algorithms Privacy in public databases Clustering with qualitative information Graph partitioning & Sparsest cut

26. Algorithms for Path-Planning Shuchi Chawla 26 Finding Bottlenecks Find the cut across which demand exceeds capacity by the largest factor 1Mbps 100Mbps 1Gbps 100Mbps Smallest but not sparsest Capacity = 201 Mbps Demand = 300 Mbps Sparsest cut Capacity is only 0.67 times the demand Sparsity of the cut = 0.67 Sparsity of a cut = capacity of cut links demand across cut 0.5Mbps

27. Algorithms for Path-Planning Shuchi Chawla 27 Finding the Sparsest Cut – Why? To help network designers –Where should more bandwidth be added –How much demand can the network support Divide-and-conquer and balanced cut To obtain good metric embeddings Negative-type metrics  Manhattan metrics Markov Chains – how fast do they mix?

28. Algorithms for Path-Planning Shuchi Chawla 28 Finding the Sparsest Cut – Results NP-hard Goal: find an approximately-sparsest cut What was known: –O(log n) approx for uniform-demands [Leighton Rao’88] unit demand between every pair of nodes based on solving an LP and rounding solution –O(log n) approx [Linial London Rabinovich ’95, Aumann Rabani ’98] –conjectured to be approximable within a constant factor –a recent breakthrough: O(  log n) approx for uniform-demands using Semi-Definite Programming [Arora Rao Vazirani ’04]

29. Algorithms for Path-Planning Shuchi Chawla 29 Finding the Sparsest Cut – Results NP-hard Goal: find an approximately-sparsest cut My results: –O(log ¾ n) approx [SODA’05: C., Gupta, R ä cke] Significant implications for partitioning and low-distortion embeddings –Evidence: NP-hard to approximate better than O(log log n) [C., Krauthgamer, Kumar, Rabani, Sivakumar] Uses an assumption in complexity theory Not a proof, but makes a constant factor approximation unlikely

30. Algorithms for Path-Planning Shuchi Chawla 30 Open Problems Closing the gap between hardness and approximation –Removing the assumption for hardness –Other techniques for approximation, e.g., based on flows Utilizing the power of SDPs –The work of Arora et al. introduced a new strong SDP formulation that was further exploited in my work Other partitioning problems –Directed networks, or one-way communication –Router capacities

31. Questions?