### Slide - University of Southern California

```Approximation Algorithms for Orienteering
and Discounted-Reward TSP
Blum, Chawla, Karger, Lane,
Meyerson, Minkoff
CS 599: Sequential Decision Making in Robotics
University of Southern California
Spring 2011
TSP: Traveling Salesperson Problem
• Graph V, E
• Find a tour (path) of shortest length that visits
each vertex in V exactly once
• Corresponding decision problem
– Given a tour of length L decide whether a tour of
length less than L exists
– NP-complete
• Highly likely that the worst case running time of
any algorithm for TSP will be exponential in |V|
Robot Navigation
• Can’t go everywhere, limits on resources
• Many practical tasks don’t require
completeness but do require immediacy or at
least some notion of timeliness/urgency (e.g.
some vertices are short-lived and need to get
to them quickly)
Prizes, Quotas and Penalties
• Prize Collecting Traveling Salesperson Problem (PCTSP)
–
–
–
–
A known prize (reward) available at each vertex
Quota: The total prize to be collected on the tour (given)
Not visiting a vertex incurs a known penalty
Minimize the total travel distance plus the total penalty, while starting
from a given vertex and collecting the pre-specified quota
– Best algorithm is a 2 approximation
• Quota TSP
– All penalties are set to zero
– Special case is k-TSP, in which all prizes are 1 (k is the quota)
– k-TSP is strongly tied to the problem of finding a tree of minimum cost
spanning any k vertices in a graph, called the k-MST problem
• Penalty TSP: no required quota, only penalties
• All these admit a budget version where a budget is given as input
and the goal is to find the largest k-TSP (or other) whose cost is no
more than the budget
Orienteering
• Orienteering: Tour with maximum possible reward
whose length is less than a pre-specified budget B
orienteering |ˌôriənˈti(ə)ri NG |noun
a competitive sport in which participants find their way to
various checkpoints across rough country with the aid of
a map and compass, the winner being the one with the
lowest elapsed time.
ORIGIN 1940s: from Swedish orientering.
Approximating Orienteering
• Any algorithm for PC-TSP extends to unrooted
Orienteering
• Thus best solution for unrooted Orienteering
is at worst a 2 approximation
• No previous algorithm for constant factor
approximation of rooted Orienteering
Discounted-Reward TSP
• Undirected weighted graph
• Edge weights represent transit time over the
edge
• Prize (reward)  v on vertex v
• Find a path visiting each vertex at time t v
t


that maximizes

v

v


Discounting and MDPs
• Encourages early reward collection, important if
conditions might change suddenly
• Optimal strategy is a policy (a mapping from states to
action)
• Markov decision process
– Goal is to maximize the expected total discounted reward
(can be solved in polynomial time) in a stochastic action
setting
– Can visit states multiple times
• Discounted-Reward TSP
– Visit a state only once (reward available only on first visit)
– Deterministic actions
Overall Strategy
• Approximate the optimum difference between
the length of a prize-collecting path and the
length of the shortest path between its endpoints
• Paper gives
– An algorithm that provably gives a constant factor
approximation for this difference
– A formula for the approximation
• The results mean that constant factor
approximations exist (and can be computed) for
Orienteering and Discounted-Reward TSP
Path Excess
• Excess of a path P from s to t: d (s,t)  d(s,t)
• Minimum excess path of total prize  is also the
minimum cost path of total prize 
• An (s,t) path approximating optimal excess  by
factor  will have 
length (by definition)
d(s,t)  


  (d(s,t)   )
• Thus a path that approximates min excess by 
cost path by 
 will also approximate minimum
P
Results
Problem
Approximation factor
Source
k-TSP
 CC
Known from prior work
(best value is 2)
Min-excess
 EP  CC 1
This paper
1  EP 
This paper
Orienteering

Discounted-RewardTSP
3
2
e( EP 1)
(roughly)
This paper

First letter is objective (cost, prize, excess, or discounted prize)
structure (path, cycle, or tree)
and second is the
Min Excess Algorithm
• Let P* be shortest path from s to t with (P * )  k
• Let (P* )  d(P* )  d(s,t)
• Min-excess algorithm returns a path P of

d(P)  d(s,t)   EP (P * )
length

with (P)  k
3
 EP  CC 1
where

2


Orienteering Algorithm
• Compute maximum-prize path of length at
most D starting at vertex s
1. Perform a binary search over (prize) values k
2. For each vertex v, compute min-excess path from
s to v collecting prize k
3. Find the maximum k such that there exists a v
where the min-excess path returned has length
at most D; return this value of k (the prize) and
the corresponding path
Discounted-Reward TSP Algorithm
1. Re-scale all edge length so   1/2
2. Replace each prize by the prize discounted by
the shortest path to that node v   d  v
3. Call this modified graph
G’

4. Guess t – the last node on optimal path P* with
excess less than 

5. Guess k – the value of (Pt* )
6. Apply min-excess approximation algorithm to
find a path P collecting scaled prize k with small
excess

7. Return this path as solution
v
Results
Problem
Approximation factor
Source
k-TSP
 CC
Known from prior work
(best value is 2)
Min-excess
 EP  CC 1
This paper
1  EP 
This paper
Orienteering

Discounted-RewardTSP
3
2
e( EP 1)
(roughly)
This paper

First letter is objective (cost, prize, excess, or discounted prize)
structure (path, cycle, or tree)
and second is the
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