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

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πv∑ γ tv

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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

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α

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d(s,t) +αε≤ α (d(s,t) +ε )

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α

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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

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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 =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

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γ=1/2

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′ π v = γd vπ v

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ε

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′ 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)