Traveling Salesman Problems Motivated by Robot Navigation

1
Traveling Salesman Problems Motivated by Robot
Navigation

• Maria Minkoff
• 
  MIT
• 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
• uncertainty in robots behavior
• battery failure
• sensor error, motor control error

3
Markov Decision Process Model

• State space S
• Choice of actions a?A at each state s
• Transition function T(ss,a)
• action determines probability distribution on
  next state
• sequence of actions produces a random path
  through graph
• Rewards R(s) on states
• If arrive in state s at time t, receive
  discounted reward gtR(s) for g?(0,1)
• MDP Goal policy for picking an action from any
  state that maximizes total discounted reward

4
Exponential Discounting

• Motivates to get to desired state 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

5
Solving MDP

• Fixing action at each state produces a Markov
  Chain with transition probabilities pvw
• Can compute expected discounted reward rv if
  start at state v
• rv rv Sw pvw gt(v,w) rw
• Choosing actions to optimize this recurrence is
  polynomial time solvable
• Linear programming
• Dynamic programming (like shortest paths)

6
Solving the wrong problem

• Package can only be delivered once
• So should not get reward each time reach target
• One solution expand state space
• New state current location ? past locations
• (packages already delivered)
• Reward nonzero only on states where current
  location not included in list of previously
  visited
• Now apply MDP algorithm
• Problem new state space has exponential size

7
Tackle an easier problem

• Problem has two novel elements for theory
• Discounting of reward based on arrival time
• Probability distribution on outcome of actions
• We will set aside second issue for now
• In practice, robot can control errors
• Even first issue by itself is hard and
  interesting
• First step towards solving whole problem

8
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)

9
Approximation Algorithms

• Discounted-Reward TSP is NP-complete
• (and so is more general MDP-type problem)
• 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

10
Related Problems

• Goal of Discounted-Reward TSP seems to be to find
  a short path that collects lots of reward
• Prize-Collecting TSP
• Given a root vertex v, find a tour containing v
  that minimizes total length foregone reward
  
  (undiscounted)
• Primal-dual 2-approximation algorithm GW 95

11
k-TSP

• Find a tour of minimum length that visits at
  least k vertices
• 2-approximation algorithm known for undirected
  graphs based on algorithm for PC-TSP Garg 99
• Can be extended to handle node-weighted version

12
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.

13
Orienteering Problem

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

14
Our Results

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

15
Our Results

• Using ?-approximation for k-TSP as subroutine
• substitute ?2 announced by Garg in 1999
• (?3/2 ?25 -approximation for Orienteering
• e(3/2 ?13-approximation for Discounted-Reward
  Collection
• constant-factor approximations for tree- and
  multiple-path versions of the problems

16
Eliminating Exponentiation

• Let dv shortest path distance (time) to v
• Define the 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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Optimum path

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

s
0
0.5

• Proof by contradiction
• Let u be first vertex with eu 1

u
1
0

• Suppose more than R/2 reward follows u

1.5
0.5

• Can shortcut directly to u then traverse
• the rest of optimum

2
1

• reduces all excesses after u by at least 1

• so undiscounts rewards by factor g -1 2

3
2

• so doubles discounted reward collected
• but this was more than R/2 contradiction

18
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-TSP
• 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

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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 S gev pv ? R/2
• ? spans prize S 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 (4 times)
  minimum excess that spans ? R/2 prize (we can
  guess R/2)
• Excesses at most 4, so gev pv ? pv/16
• ? discounted reward on found path ? R/32

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Solving Min-Excess Path problem

• Exactly solvable case monotonic paths
• Suppose optimum path goes through vertices in
  strictly increasing distance from root
• Then can find optimum by dynamic program
• Just as can solve longest path in an acyclic
  graph
• Build table
• For each vertex v is there a monotonic path from
  v with length l and prize p?

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Solving Min-Excess Path problem

• Approximable case wiggly paths
• Length of path to v is lv dv ev
• If ev gt dv then lv gt ev gt lv/2
• i.e., take twice as long as necessary to reach v
• So if approximate lv to constant factor, also
  approximate ev to twice that constant factor

22
Approximating path length

• Can use k-TSP algorithm to find approximately
  shortest s-t path with specified prize

• merge s and t into vertex r
• opt path becomes a tour
• solve k-TSP with root r

s
t

• unmerge can get one or more cycles

• connect s and t by shortest path

23
Decompose optimum path
monotone
monotone
monotone
wiggly
wiggly
Divides into independent problems
gt 2/3 of each wiggly path is excess
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Decomposition Analysis

• 2/3 of each wiggly segment is excess
• That excess accumulates into whole path
• total excess of wiggly segment ? excess of whole
  path
• total length of wiggly segments ? 3/2 of path
  excess
• Use dynamic program to find shortest (min-excess)
  monotonic segments collecting target prize
• Use k-TSP to find approximately shortest wiggles
  collecting target prize
• Approximates length, so approximates excess
• Over all monotonic and wiggly segments,
  approximates total excess

25
Dynamic program for Min-Excess Path

• For each pair of vertices and each (discretized)
  prize value, find
• Shortest monotonic path collecting desired prize
• Approximately shortest wiggly path collecting
  desired prize
• Note polynomially many subproblems
• Use dynamic programming to find optimum pasting
  together of segments

26
Solving Orienteering Problem special case
s

• Given a path from s that
• collects prize P
• has length ? D
• ends at t, the farthest point from s

0
0.5
1

• For any const integer r ? 1, there
• exists a path from s to some v with
• prize ? P/r
• excess ? (D-dv)/r

1.5
2
1
v
3
t
27
Solving Orienteering Problem
s

• General case path ends at arbitrary t
• Let u be the farthest point from s
• Connect t to s via shortest path
• One of path segments ending at u
• has prize ? P/2
• has length ? D
• ? Reduced to special case
• Using 4-approximation for Min-Excess Path get
  8-approximation for Orienteering

t
u
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Budget Prize-Collecting Steiner Tree problem

• Find a rooted tree of edge cost at most D that
  spans maximum amount of prize
• Complement of k-MST
• Create Euler tour of opt tree T of cost ? 2D
• Divide this tour into two paths starting at root
  each of length ? D
• One of them contains at least ½ of total prize
• Path is a type of tree
• Use c-approximation algorithm for Orienteering to
  obtain 2c-approximation for Budget PCST

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Summary

• Showed maximum discounted reward can be
  approximated using min-excess path
• Showed how to approximate min-excess pathusing
  k-TSP
• Min-excess path can also be used to solve rooted
  Orienteering problem (open question)
• Also solves tree and cycle versions of
  Orienteering

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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
• Directed graphs
• We used k-TSP, only solved for undirected
• For directed, even standard TSP has no known
  constant factor approximation
• We only use k-TSP/undirectedness in wiggly parts

31
Future directions

• Stochastic actions
• Stochastic seems to imply directed
• Special case forget rewards.
• Given choice of actions, choose to minimize cover
  time of graph
• Applying discounting framework to other problems
  
• Scheduling
• Exponential penalty in place of hard deadlines