Approximation Algorithms for Path-Planning Problems

1. Approximation Algorithms for Path-Planning Problems with Nikhil Bansal, Avrim Blum and Adam Meyerson Shuchi Chawla

2. The Trick-o-Treaters Problem • Collect as much candy as possible within 6pm and 8pm • More candy  more popularity with the kids • Some complicating constraints • Limited amount of time • Mr. X always gives twice as much candy as Mrs. Y, but his house is a long detour. • Orienteering: Given a metric and a starting point, cover as many “high-reward” locations as possible within a limited amount of time Shuchi Chawla, Carnegie Mellon University

3. Path-planning in the real world • A robot-navigation problem • Deliver packages to certain locations • Faster delivery => greater happiness • Limited battery power • Packages have different deadlines for delivery • Assembly analysis • Manufacturing • Production planning Shuchi Chawla, Carnegie Mellon University

4. A reward-time trade-off • Given graph (metric) G, construct a path satisfying some constraints and optimizing some function. • Classic formulation – Traveling Salesman Find the shortest tour covering all locations • Budget the path-length and maximize reward • Orienteering Hard bound on path length • Time Window Visit node v within [Rv, Dv] • Impose a reward quota and minimize length • k-Path Collect at least k reward Shuchi Chawla, Carnegie Mellon University

5. A reward-time trade-off • Given graph (metric) G, construct a path satisfying some constraints and optimizing some function. • Classic formulation – Traveling Salesman Find the shortest tour covering all locations • Budget the path-length and maximize reward • Orienteering4 [Blum C Karger+03] 3 [Bansal Blum C Meyerson 04] • Time Window3log2n [Bansal Blum C Meyerson 04] • Impose a reward quota and minimize length • k-Path 2 +  [Chaudhury Godfrey Rao+ 03] Shuchi Chawla, Carnegie Mellon University

6. The rest of this talk • A 3-approximation for Orienteering • An O(log2n) approx for the Time-Window Problem • Orienteering with deadlines • Incorporating release-dates • Extensions and Open Problems Shuchi Chawla, Carnegie Mellon University

7. Orienteering and k-Path • Orienteering : length · D; maximize reward • k-Path : reward ¸ k ; minimize length • Complementary problems • Series of results on k-TSP (related to k-Path) [BRV99] [Garg99] [AK00] [CGRT03] … best approx: (2+) • None for Orienteering until recently! Shuchi Chawla, Carnegie Mellon University

8. Why is Orienteering difficult? • 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! OPT(d) s APPROX Shuchi Chawla, Carnegie Mellon University

9. Why is Orienteering difficult? • First attempt – Use distance-based approximations to approximate reward • Idea – Modify the algorithm itself • Doesn’t help – moat-growing always goes for shallow fruit • Orienteering is inherently harder; Perturbation of the input changes the output widely OPT(d) s APPROX Shuchi Chawla, Carnegie Mellon University

10. Why is Orienteering difficult? t s • Second attempt – approximate subparts of the optimal path and shortcut other parts • If we stray away from the optimal path by a lot, we may not be able to cover reward that’s far away • Approximate the “extra” length taken by a path over the shortest path length OPT APPROX Shuchi Chawla, Carnegie Mellon University

11. Why is Orienteering difficult? Min-Excess Path Problem • Second attempt – approximate subparts of the optimal path and shortcut other parts • If we stray away from the optimal path by a lot, we may not be able to cover reward that’s far away • 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

12. The Min-Excess Problem • Given graph G, start and end nodes s, t, reward on nodes v,target reward k, find a path that collects reward at least k and minimizes(P) =ℓ(P) – d(s,t) • At optimality, this is exactly the same as the k-path objective of minimizing ℓ(P) • However, approximation is different: Min-excess is strictly harder than K-path • There is a (2+)-approximation for Min-Excess [Blum, C, Karger, Meyerson, Minkoff, Lane, FOCS’03] • Our algorithm returns a path with length d(s,t) + (2+) (P) excess Shuchi Chawla, Carnegie Mellon University

13. A 3-approximation to Orienteering t 3 s 1 2 • There exists a path from s to t, that • collects reward at least  • has length  D • Given a 3-approximation to min-excess: 1. Divide into 3 “equal-reward” parts (hypothetically) 2. Approximate the part with the smallest excess  3-approximation to orienteering • Using an r-approx for Min-excess ( r Z+ ), we get an r-approximation for s-t Orienteering Excess of path P (P) = dP(u,v)– d(u,v) Open: Given an r-approx for min-excess (r 2R +), can we get r-approx to Orienteering? v2 OPT v1 APPROX Excess of one path · (1+2+3)/3 Can afford an excess up to (1+2+3) Shuchi Chawla, Carnegie Mellon University

14. So far… • A 3-approximation for Orienteering • An O(log2n) approx for the Time-Window Problem • Orienteering with deadlines • Incorporating release-dates • Extensions and Open Problems Coming up… Shuchi Chawla, Carnegie Mellon University

15. The Time-Window Problem • Find a path visiting many nodes in their time-window school bus routing bank and postal deliveries industrial refuse collection newspaper delivery fuel oil delivery dial-a-ride service • Widely studied in scheduling and OR literature • Constant-approx known for points on a line, few different time-windows; No approximation known for the general case • A special case – The Deadline-TSP Problem • Vertices only have deadlines • All “release-times” are 0. Shuchi Chawla, Carnegie Mellon University

16. The next step: Deadline-TSP • Every vertex has a deadline D(v); Find a path that maximizes nodes v visited before D(v) • If the last node on the path has the min deadline, use Orienteering to approximate the reward • Everything visited before the minimum deadline • Don’t need to bother about deadlines of other nodes • Does OPT always have a large subpath with the above property? • There are many subpaths of OPT with the above property that together contain all the reward NO! Shuchi Chawla, Carnegie Mellon University

17. A segmentation of OPT Deadline Time Shuchi Chawla, Carnegie Mellon University

18. Deadline-TSP • Segment graph into many parts, approximate each using Orienteering and patch them together • How do we find such a segmentation without knowing the optimal path? • In order to avoid double-counting of reward, segments should be node-disjoint • Our result – There exists a segmentation based only on deadlines, such that the resulting solution is a (3 log n)-approximation Shuchi Chawla, Carnegie Mellon University

19. A 2-dimensional view minimal vertices Deadline “Disjoint Rectangles” Time Shuchi Chawla, Carnegie Mellon University

20. The Rectangle Argument • Approximate reward contained in a “disjoint” family of rectangles • Every pair of rectangles is non-overlapping in BOTH dimensions • We construct O(log n) families of disjoint rectangles 1. These cover ALL the reward in OPT 2. We can approximate the best of them • We get an O(log n)-approximation Shuchi Chawla, Carnegie Mellon University

21. The Rectangle Argument Deadline Time • There are O(log n) families of disjoint rectangles that cover all the reward in OPT Shuchi Chawla, Carnegie Mellon University

22. The Rectangle Argument If there are between 2b and 2b+1 points in between, then either the bth or a larger family contains exactly 1 point in the interval • There are O(log n) families of disjoint rectangles that cover all the reward in OPT Deadline Time Shuchi Chawla, Carnegie Mellon University

23. The Rectangle Argument 2. We can approximate the best disjoint family • Suppose we know the minimal vertices • Just try out all the log n families • Problem - Minimal vertices depend on the optimal tour! • Solution – Try all possibilities. They are ordered by deadlines, so use a simple dynamic program Shuchi Chawla, Carnegie Mellon University

24. The Rectangle Argument Deadline Time 2. We can approximate the best disjoint family Shuchi Chawla, Carnegie Mellon University

25. The O(log n)-approximation • Approximate reward contained in a “disjoint” family of rectangles • Every pair of rectangles is non-overlapping in BOTH dimensions • We construct O(log n) families of disjoint rectangles 1. These cover ALL the reward in OPT 2. We can approximate the best of them • Obtain an O(log n)-approximation Shuchi Chawla, Carnegie Mellon University

26. From Deadlines to Time-Windows t s s t • Nodes have deadlines as well as release times • Note that release times are dual to deadlines – if we look at the path from the end to the start, release times become deadlines! • Log-approximation for deadlines  log-approximation for release dates • Algorithm for Time-Windows: • Run the approximation for Deadline-TSP • Replace Orienteering by Orienteering with release-dates • O(log2n)-approximation for the Time-Window problem ℓ(OPT) = L D(v) = L-R(v) OPT v Require ℓ(s,v)  R(v)  ℓ(t,v)  L-R(v) Shuchi Chawla, Carnegie Mellon University

27. A Bicriteria Approximation • Given any  > 0, Get O(log 1/) fraction of reward Exceed deadlines by a (1+) factor • O( log Dmax )-approximation • Constant factor approximation if we can exceed deadlines by a small constant factor • Nice trade-off: Halving the extra time taken, increases the approximation factor by only an additive 1 Shuchi Chawla, Carnegie Mellon University

28. An overview of our results Problem Approximation Orienteering 3 Deadline TSP 3 logn Time-Window Problem 3 log2n reward: log 1/ deadlines: 1+ Time-Window Problem - bicriteria Shuchi Chawla, Carnegie Mellon University

29. Future Directions • Better approximations • can we get a constant factor for Time-Windows? • special metrics such as trees or planar graphs • hardness of approximation? • Asymmetric Path-planning • the graph is directed; still obeys triangle inequality • polylog-approximations and lower bounds for distance • need entirely different ideas for asymmetric-Orienteering • is it log-hard? Shuchi Chawla, Carnegie Mellon University

30. Questions? Shuchi Chawla, Carnegie Mellon University