Approximation Algorithms for Path-Planning Problems Nikhil Bansal, Avrim Blum, Shuchi Chawla and Adam Meyerson Carnegie Mellon University

Shuchi Chawla, Carnegie Mellon University 2 The FEDEX Problem  Several packages to be delivered  Each has a location, priority value, delivery window  Problem: Deliver as many packages as possible Must deliver within its delivery window, or, time-window May not be able to deliver all Maximize number delivered Higher priority receives more importance Maximize total “reward”

Shuchi Chawla, Carnegie Mellon University 3 The Time-Window Problem  Given a metric space G and a starting point r  Find a path visiting many nodes in their time-window  Objective: maximize total reward on nodes visited  package delivery  school bus routing  dial-a-ride service  newspaper delivery

Shuchi Chawla, Carnegie Mellon University 4 Some History  NP-hard even on a line [Tsitsiklis ’92]  Widely studied in scheduling and OR literature [ Kolen et al ’87 ] [ Desrochers et al ’92 ] [ Kantor et al ’92 ] [ Thengiah et al ’95 ] [ Tan et al ’00 ] Optimal algorithm for points on line with no release times [Tsitsiklis ’92] O(log n)-approx on a line [Bar-Yehuda Even Shahar ’03] Constant approx for constant number of time-windows [Chekuri Kumar ’04]  No general case approximation known previously

Shuchi Chawla, Carnegie Mellon University 5 Special cases  Closely related to the Orienteering problem All vertices have release-times = 0, and deadline = D Visit as many vertices as possible by time D [Blum et. al. ’03] gave a 4-approximation This paper: 3-approximation  A special case – The Deadline-TSP Problem Vertices only have deadlines All “release-times” are 0

Shuchi Chawla, Carnegie Mellon University 6 Deadline TSP3 log n An overview of our results Time-Window Problem3 log 2 n ApproximationProblem Orienteering3 Time-Window Problem - bicriteria reward: log 1/  deadlines: 1+  This talk

Shuchi Chawla, Carnegie Mellon University 7 Approximating 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 8 A segmentation of OPT Time Deadline

Shuchi Chawla, Carnegie Mellon University 9 Approximating 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 10 A 2-dimensional view Time Deadline minimal vertices “Disjoint Rectangles” Segment nodes into disjoint rectangles deadlinetime and

Shuchi Chawla, Carnegie Mellon University 11 The Rectangle Argument  Approximate reward contained in a family of disjoint rectangles Every pair of rectangles is non-overlapping in BOTH dimensions  We construct  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 12 The Rectangle Argument 1.There are  log n families of disjoint rectangles that cover all the reward in OPT Time Deadline

Shuchi Chawla, Carnegie Mellon University 13 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, permitting a simple dynamic program (Details omitted)

Shuchi Chawla, Carnegie Mellon University 14 From Deadlines to Time-Windows  Nodes have deadlines as well as release times  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  O(log 2 n)-approximation for the Time-Window problem

Shuchi Chawla, Carnegie Mellon University 15 A Bicriteria Approximation  Given any  > 0, Get O(log 1/  ) fraction of reward Exceed deadlines by a (1+  ) factor  O( log D max )-approximation  Constant factor approximation if we can exceed deadlines by a small constant factor

Shuchi Chawla, Carnegie Mellon University 16 Future Directions  Better/faster approximations constant factor for Time-Windows? special metrics such as trees or planar graphs  Hardness of approximation log-hardness for Time-Windows?  Asymmetric / Online versions

Shuchi Chawla, Carnegie Mellon University 17 Questions?