approximation algorithms for orienteering and discounted-reward tsp blum, chawla, karger, lane,...
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Approximation Algorithms for Orienteering and Discounted-Reward TSP
Blum, Chawla, Karger, Lane, Meyerson, Minkoff
CS 599: Sequential Decision Making in RoboticsUniversity 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
orienteering |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) on vertex v• Find a path visiting each vertex at time
that maximizes
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πv
€
πv∑ γ tv
€
tv
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:• 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)
• Thus a path that approximates min excess by will also approximate minimum cost path by
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dP (s, t) − d(s, t)
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Π
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Π
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ε
€
α
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d(s, t) +αε
≤ α (d(s, t) +ε )
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α
€
α
ResultsProblem Approximation factor Source
k-TSP Known from prior work (best value is 2)
Min-excess This paper
Orienteering This paper
Discounted-Reward TSP (roughly) This paper€
αCC
€
αEP =3
2α CC +1
€
1+ α EP⎡ ⎤
€
e(α EP +1)
First letter is objective (cost, prize, excess, or discounted prize)and second is the structure (path, cycle, or tree)
Min Excess Algorithm
• Let P* be shortest path from s to t with • Let • Min-excess algorithm returns a path P of
length withwhere
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Π(P*) ≥ k
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ε(P*) = d(P*) − d(s, t)
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d(P) = d(s, t) +α EPε (P*)
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Π(P) ≥ k
€
αEP =3
2α CC +1
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 k3. 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 2. Replace each prize by the prize discounted by the
shortest path to that node3. 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 6. Apply min-excess approximation algorithm to find
a path P collecting scaled prize k with small excess7. Return this path as solution
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γ=1/2
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′ π v = γd vπ v
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ε
€
′ Π (Pt*)
ResultsProblem Approximation factor Source
k-TSP Known from prior work (best value is 2)
Min-excess This paper
Orienteering This paper
Discounted-Reward TSP (roughly) This paper€
αCC
€
αEP =3
2α CC +1
€
1+ α EP⎡ ⎤
€
e(α EP +1)
First letter is objective (cost, prize, excess, or discounted prize)and second is the structure (path, cycle, or tree)