Traveling Salesman Problems Motivated by Robot Navigation

1
Traveling Salesman Problems Motivated by Robot
Navigation

• based on joint work with Avrim Blum, Shuchi
  Chawla, David Karger, Terran Lane,
• Adam Meyerson

2
A Robot Navigation Problem

• Robot delivering packages in a building
• Goal to deliver as quickly as possible
• Classic model Traveling Salesman Problem
• find a tour of minimum length
• Additional constraints
• some packages have higher priority
• robots lifetime is uncertain
• power loss
• catastrophic failure

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Exponential Discounting

• Markov Decision Process approach
• assign reward ri to package i
• robot receives discounted reward gt ri
• for delivering package i at time t
• Motivates to deliver high-priority packages
  quickly
• Inflation reward collected in distant future
  decreases in value due to uncertainty
• at time t robot loses power with fixed
  probability
• probability of being alive at t is exponentially
  distributed
• discounting reflects value of reward in
  expectation

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Discounted-Reward TSP

• Given
• undirected graph G(V,E)
• edge weights (travel times) de 0
• weights on nodes (rewards) rv 0
• discount factor ? ? (0,1)
• root node s
• Goal
• find a path P starting at s that maximizes
• total discounted reward ?(P) ?v?P rv ?dP(v)

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Approximation Algorithms

• Discounted-Reward TSP is NP-hard
• reduction from minimum latency TSP
• So intractable to solve exactly
• Goal approximation algorithm
• that is guaranteed to collect at least some
  constant fraction of the best possible discounted
  reward

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Related Problems

• Goal of Discounted-Reward TSP seems to be to find
  a short path that collects lots of reward
• k-TSP and k-path
• Find a tour (path) of minimum length that starts
  at a given node s and visits at least k vertices
• (2?)-approximation algorithm for k-TSP AK00
• (2?)-approximation algorithm for k-path follows
  from work of Chaudhuri et al. CGRT03
• Mismatch constant factor approximation on length
  doesnt exponentiate well

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Mismatch

• Constant factor approximation on length doesnt
  exponentiate well
• Suppose optimum solution reaches some vertex v at
  time t for reward gtr
• Constant factor approximation would reach within
  time 2t for reward g2tr
• Result get only gt fraction of optimum
  discounted reward, not a constant fraction.

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Orienteering Problem

• Find a path of length at most D that maximizes
  net reward collected
• Complement of k-path
• approximates reward collected instead of length,
  so exponentiation doesnt hurt
• unrooted case can be solved via k-TSP or k-path
• Drawback
• no constant factor approximation for rooted
  non-geometric version previously known

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Optimizing for expected outcome

• Orienteering can be used to optimize reward
  collected in expected lifetime
• Discounted-Reward TSP optimizes expected reward
  collected over robots lifetime
• Ereward ? reward collected in Etime

s
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Our Results

• Using ?-approximation for k-path as subroutine
• ?3/2 ?1/2? -approximation for Orienteering
• e(3/2 ? 1/2)-approximation for
  Discounted-Reward TSP
• constant-factor approximations for tree- and
  multiple-path versions of the problems

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Our Results

• Using ?-approximation for k-path as subroutine
• substitute ? 2? from Chaudhuri et al.
• ?3/2 ?1/4?-approximation for Orienteering
• e(3/2 ? 8.5 -approximation for
  Discounted-Reward TSP
• constant-factor approximations for tree- and
  multiple-path versions of the problems

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Eliminating Exponentiation

• Let dv shortest path distance (time) to v
• Define prize at v as pvgdv rv
• max discounted reward possibly collectable at v
• If given path reaches v at time tv,
• define excess ev tv dv
• difference between shortest path and chosen one
• Then discounted reward at v is gev pv
• Idea if excess small, prize discounted reward
• Fact excess only increases as traverse path
• excess reflects lost time cant make it up

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Property of optimum path

• assume g ½ (can scale edge lengths)
• Claim at least 1/2 of optimum paths discounted
  reward R is collected before paths excess
  reaches 1

s
0
0.5

• Proof shortcut to u
• reduces all excesses after u by at least 1
• so undiscounts rewards by factor g-12
• so doubles discounted reward collected

u
1
0
1.5
0.5
2
1

• Algorithm idea
• Guess u
• Find a path to u of small excess that spans ?
  R/2 prize

3
2
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New problem Approximate Min-Excess Path

• Suppose there exists an s-t path P with prize
  value ? of length l(P)dte
• Optimization find s-t path P with prize value
  ? that minimizes excess l(P)-dt over shortest
  path to t
• equivalent to minimizing total length, e.g.
  k-path
• Approximation find s-t path P with prize value
  ? that approximates optimum excess over shortest
  path to t, i.e. has length l(P) dt ce
• better than approximating entire path length
• obtain (2.5?)-approximation algorithm via
  dynamic programming using k-path algorithm

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Decompose optimum path
distance from s
s
t
monotone
monotone
monotone
wiggly
wiggly

• Divides into independent problems
• exactly solvable case monotonic paths
• approximable case wiggly paths
• gt 2/3 of each wiggly path is excess
• approximate excess by approximating length

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Using Min-Excess Path

• Recall discounted reward at v is gev pv
• Prefix of optimum discounted reward path
• collects discounted reward ? gev pv ? R/2
• ? spans prize ? pv ? R/2
• and has no vertex with excess over 1
• Guess t last node on opt path with excess et ?
  1
• Find a path to t of approximately (3 times)
  minimum excess that spans ? R/2 prize (we can
  guess R/2)
• Excesses at most 3, so gev pv ? pv/8
• ? discounted reward on found path ? R/16

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Summary

• Algorithm for approximately maximizing Ereward
  over uncertain lifetime
• Our techniques also give 4-approximation for
  previously open Orienteering problem of
  maximizing reward over fixed period of time
• Open questions
• non-uniform discount factors
• each vertex v has its own ?v
• non-uniform deadlines
• each vertex specifies its own deadline by which
  it has to be visited in order to collect reward
• d irected graphs