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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 - PowerPoint PPT Presentation
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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
€
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
€
dP (s,t) − d(s, t)
€
Π
€
Π
€
ε
€
α
€
d(s,t) +αε≤ α (d(s,t) +ε )
€
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 =32α 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
€
Π(P*) ≥ k
€
ε(P*) = d(P*) − d(s,t)
€
d(P) = d(s, t) +α EPε (P*)
€
Π(P) ≥ k
€
αEP =32α 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
€
γ=1/2
€
′ π v = γd vπ v
€
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 =32α 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)