Algorithms for Path-Planning

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Algorithms for Path-Planning

• Shuchi Chawla
• Carnegie Mellon / Stanford Univ.

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A trick-or-treat problem

• Its Halloween
• Collect large amounts of candy between 6pm and
  8pm
• Goal Get as much candy as possible
• In what order should you visit houses?

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A travelling repair-bot problem

• Robot receives requests for repair
• Requests come with a time-window for servicing
• Brownie points for each request serviced
• Goal Maximize the total brownie points
• Cannot perform all of them
• Takes time to service each request and move from
  one to another
• The problem
• Which ones to accept?
• How to schedule them?

Selection
Ordering
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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
• Delivery distribution problems
• Production planning, Supply chain management
• Robot navigation
• Mostly NP-hard ?

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Path-planning A brief history

• Studied in Operations Research for the past 20-30
  years
• (also known as Vehicle Routing)
• Many variants
• multiple vehicles, stochastic demands,
  pickup-and-delivery
• Techniques for Mixed Integer Programming
• Stochastic planning problems in Robotics and AI
• My focus Approximation algorithms

A 2-approximation half-as-good as OPT
value ? ½ OPT
Find a solution in polynomial-time with value ?
OPT (a ? 1)
1/?

• cutting plane methods ? branch and bound
• simulated annealing ? genetic
  algorithms

• Focus mostly on exploration, position
  estimation
• Some poly-time solvable problems studied e.g.
  shortest paths
• Linear and Dynamic programming techniques

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Approximation Results

• A reward vs. time trade-off

A quota approximation
OPT
(2 hrs, )
(4 hrs, )
A budget approximation
(2 hrs, )
Reward obtained
Time taken
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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 locations visit k locations visit k
locations, but minimize excess
minimize time plus reward foregone max. reward,
reward decreases with time
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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

This talk
? ? ?
3 3 log n 3 log2 n
Use structural properties Dynamic
Programming FOCS03 STOC04
Use LP-rounding
1.5 Christofides 76 2 BRV99 AK00
CGRT03 Garg05 2?
1.5 Christofides 76 2 BRV99 AK00
CGRT03 Garg05 ?
2 Goemans Williamson 92 ?
2 Goemans Williamson 92 6.75?
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Back to candy-collection (Orienteering)

• The givens
• A map G of locations, distances, start
  location s
• Rewards ? on locations
• Deadline D
• To find
• A path that collects as much reward as possible
  by deadline D
• NP-hard we want to find an approximation

In poly-time, find a path covering reward ? ? OPT
by deadline D
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Back to Orienteering why LPs dont work

• Naïve attempt use (a variant of) k-TSP
• If there exists a path length , reward ?
• 2-approx to k-TSP gives length ? , reward ?

½D
1/?
OPT
D
OPT
1/?
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Back to Orienteering why LPs dont work

• Naïve attempt use (a variant of) k-TSP
• Fails when most reward is far from the start
• Algorithms for quota problems rely on the
  Goemans-Williamson primal-dual subroutine
• Miss out on far-away reward
• Budget problems are ill-behaved w.r.t. small
  perturbations
• Solution handle the very-low slack case
  separately

Bad case low slack
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Formalizing the slack

• Excess of path length of path length of
  shortest path
• The Min-Excess problem
• Given a map G, start s, end t, reward quota k
• Find path with reward k that minimizes excess

t
dist(s,t)
1½ hrs.
length
2 hrs.
OPT
excess
l
length shortest distance
½ hr.
Min-Excess approx
2½ hrs.
k-TSP approx
4 hrs.
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Formalizing the slack

• Excess of path length of path length of
  shortest path
• An approx. for excess ? an approx. for
  Orienteering

• a-approx to length len(ALG) ? a (shorest-len
  excess)
• a-approx to excess len(ALG) ? shortest-len a
  excess

Given approx for k-TSP Good solution to prefix
of OPT
Given approx for excess Good solution to
suffix of OPT
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Approximating the excess

• Excess of path length of path length of
  shortest path
• a-approx to length len(ALG) ? a (shortest-len
  excess)
• a-approx to excess len(ALG) ? shortest-len a
  excess
• The simple case excess gt shortest-len
• 2-approx to length gives 3-approx to excess
• Solution use k-TSP
• The harder case excess ltlt shortest-len

• examine a slightly different problem
• Suppose OPT visits all nodes in order of
  increasing distance from start Find OPT.
• Can solve this exactly using dynamic programming!

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Approximating the excess combining them

• The large excess case
• excess gt shortest-len
• Use k-TSP

• The small excess case
• excess 0
• (OPT is monotone)
• Use Dynamic Programming

Gives a (2?)-approximation for Min-Excess
FOCS03
Patch segments using dynamic programming
What about the intermediate case?
OPT
wiggly
wiggly
monotone
monotone
monotone
In order of increasing distance from start
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An algorithm for Orienteering

• 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 OPT into 3 equal-reward parts
  (hypothetically)
• 2. Approximate the part with the smallest excess
• ? 3-approximation to orienteering

An r-approximation for min-excess gives a
?r?-approximation for Orienteering
Excess of path from u to v ?(u,v) l(u,v)d(u,v)
OPT
v2
v1
ALG
Excess of ALG ? ?1?2?3 Reward of ALG ?
reward of OPT
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So far

• A new problem min-excess
• An approximation to min-excess
• A 3-approximation to Orienteering
• Coming up next
• Planning with deadlines a (3 log
  n)-approximation

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Planning with deadlines Deadline-TSP

• The travelling repair-bot problem
• Every node (request) v has a deadline D(v)
• Reward obtained at v, if v is visited before D(v)
• In Orienteering, D(v) D for all nodes v
• Constant approx known for special cases
• points on a line
• Few different deadlines (based on Orienteering)
• Nothing known in general

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Approximating Deadline-TSP

• Our approach
• Divide the problem into Orienteering
  sub-problems
• Solve these using algorithm developed previously
• When can we use Orienteering?
• If the last vertex visited by OPT has the
  smallest deadline
• Remove nodes of smaller deadline
• Reduce all other deadlines to the smallest one
• OPT remains unchanged
• Does OPT have a large subpath with this property?

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A pictorial representation of OPT
Good rectangle
Deadline
Can be approximated via Orienteering
Time
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The high-level idea

• A single good rectangle may not contain enough
  reward
• Idea Approximate many rectangles stitch
  together the approximations
• Main bottleneck avoiding double-counting of
  reward
• Divide graph into disjoint subsets Vi
• Approximate the i-th rectangle over Vi

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Avoiding double-counting

• Divide nodes into sets by deadlines
• Guess last node on OPT in each group
• Approximate Orienteering on each group

V4
V3
V2
Deadline
V1
length of seg. 2
length of seg. 3
Time
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Avoiding double-counting

• To visit nodes before deadlines, rectangles must
  be disjoint along the time-axis
• Still consider the same sets Vi of vertices
• Obtain reward contained in axis-disjoint
  rectangles

V4
V3
V2
Deadline
V1
Time
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The disjoint-rectangle argument

• Approximate reward contained in a family of
  disjoint rectangles
• We construct (log n) families of disjoint
  rectangles
• The families together contain all the nodes of
  OPT
• We can approximate the best family via
  Orienteering and dynamic programming
• Reward obtained ? ? 1/(log n)
• Gives a (3 log n)-approximation

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Deadlines to Time-Windows

• Nodes have release-times as well as deadlines
• Must visit a node within its time-window to
  obtain reward
• Straightforward extension of Deadline-TSP
  algorithm
• ? (3 log2 n)-approximation to Time-Windows-TSP
• A bicriteria approximation
• Given any k gt 0, get O(1/k) fraction of reward
• exceed deadlines by a (12-k) factor
• ? O(log Dmax)-approximation

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A recap of our results

• First approximations for Orienteering,
  Deadline-TSP, Time-Windows-TSP,
  Discounted-Reward-TSP,
• Introducing the new min-excess objective and
  approximating it

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Open Problems

• Hardness of planning problems
• Approximations for directed path-planning
• Chekuri, Pal give quasi-polytime
  polylog-approximations
• Dealing with capacity precedence constraints
• Techniques for approximating planning problems on
  MDPs

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Questions?