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

Shuchi Chawla, Carnegie Mellon University 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 [R v, D v ]  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 Orienteering 4 [Blum C Karger+03] 3 [Bansal Blum C Meyerson 04] Time Window 3log 2 n [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(log 2 n) 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! s OPT(d) 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 s OPT(d) APPROX

Shuchi Chawla, Carnegie Mellon University 10 Why is Orienteering difficult?  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 s t OPT APPROX

Shuchi Chawla, Carnegie Mellon University 11 Why is Orienteering difficult?  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+  Min-Excess Path Problem

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  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 s t Excess of one path · (  1 +  2 +  3 )/ 3 Can afford an excess up to (  1 +  2 +  3 ) 11 22 33 Excess of path P  (P) = d P (u,v)– d(u,v)  Using an r-approx for Min-excess ( r  Z + ), we get an r-approximation for s-t Orienteering v1v1 v2v2 OPT APPROX Open: Given an r-approx for min-excess (r 2 R + ), can we get r-approx to Orienteering?

Shuchi Chawla, Carnegie Mellon University 14 So far…  A 3-approximation for Orienteering  An O(log 2 n) 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 Time Deadline

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 Time Deadline minimal vertices “Disjoint Rectangles”

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 1.There are O(log n) families of disjoint rectangles that cover all the reward in OPT Time Deadline

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

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 2. We can approximate the best disjoint family Time Deadline

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  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(log 2 n)-approximation for the Time-Window problem s t OPT ℓ(OPT) = L v Require ℓ(s,v)  R(v)  ℓ(t,v)  L-R(v) st D(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 D max )-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 Deadline TSP3 logn An overview of our results Time-Window Problem3 log 2 n ApproximationProblem Orienteering3 Time-Window Problem - bicriteria reward: log 1/  deadlines: 1+ 

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 31 The Min-Excess Problem  Given graph G, start and end nodes s, t, reward on nodes  v  Find a path from s to t collecting K reward and minimizing 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  -approx to K-path :  ℓ (P)  -approx to min-excess : d +  ( ℓ (P) – d) =  ℓ (P) – (  -1)d  Min-excess is strictly harder than K-path

Shuchi Chawla, Carnegie Mellon University 32 Solving Min-Excess  OPT = d+  ; k-path gives us ALG =  (d+  ) We want ALG = d +   Note: When  ≈ d,  (d+  ) ≈ d + 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 33  OPT = d+  ; k-path gives us ALG =  (d+  ) We want ALG = d +   Note: When  ≈ d,  (d+  ) ≈ d + 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 to solve exactly! Solving Min-Excess  Binary decision for each vertex – should it be in the path or not?  Compute P(v j,t) = the “best” path that has length t and ends at v j  P(v j+1,t) == consider P(u,t’), where t’ = t- ℓ (u,v j+1 ) pick the best path (best u) from the above

Shuchi Chawla, Carnegie Mellon University 34  Idea: When  is large, approximate using k-path  Idea: Completely monotone path  use dynamic programming to solve exactly! Solving Min-Excess monotone wiggly s t Approximate Dynamic Program Patch segments using dynamic programming OPT