Approximation Algorithms for PathPlanning Problems

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Approximation Algorithms for Path-Planning
Problems

• Nikhil Bansal, Avrim Blum, Shuchi Chawla and Adam
  Meyerson

Carnegie Mellon University
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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

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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

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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

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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

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An overview of our results
Approximation
Problem
Orienteering
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This talk
Deadline TSP
3 log n
Time-Window Problem
3 log2n
reward log 1/? deadlines 1?
Time-Window Problem - bicriteria
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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
• Dont 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!
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A segmentation of OPT
Deadline
Time
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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

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A 2-dimensional view
Disjoint Rectangles
Deadline
deadline
time and
Segment nodes into disjoint rectangles
Time
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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

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The Rectangle Argument

• There are ? log n families of disjoint rectangles
  that cover all the reward in OPT

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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)

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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(log2n)-approximation for the Time-Window problem

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A Bicriteria Approximation

• Given any ? gt 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

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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

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Questions?