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
robot’s lifetime is uncertain
power loss
catastrophic failure

3. 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

4. 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)

5. 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

6. 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 [AK’00]
(2+)-approximation algorithm for k-path follows from work of Chaudhuri et al. [CGRT’03]
Mismatch: constant factor approximation on length doesn’t exponentiate well

7. 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 doesn’t hurt
unrooted case can be solved via k-TSP or k-path
Drawback
no constant factor approximation for rooted non-geometric version previously known

8. 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

9. 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  approximation for Discounted-Reward TSP
constant-factor approximations for tree- and multiple-path versions of the problems

10. Eliminating Exponentiation
Let dv = shortest path distance (time) to v
Define prize at v as pv=gdv 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; can’t make it up

11. Property of optimum path
assume g = ½ (can scale edge lengths)
Claim: at least 1/2 of optimum path’s discounted reward R is collected before path’s excess reaches 1 s
0.5
Proof : shortcut to u
reduces all excesses after u by at least 1
so “undiscounts” rewards by factor g-1=2
so doubles discounted reward collected u 1
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

12. New problem: Approximate Min-Excess Path
Suppose there exists an s-t path P* with prize value  of length l(P*)=dt+e
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
Remind what k-TSP is

13. 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
> 2/3 of each wiggly path is excess
approximate excess by approximating length

14. Summary
Algorithm for approximately maximizing E[reward] 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