Chandra Chekuri, Nitish Korula and Martin Pal Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms (SODA 08) Improved Algorithms.

Chandra Chekuri, Nitish Korula and Martin Pal

Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms (SODA 08)

Improved Algorithms for Orienteering and Related Problems

Presented By: Asish Ghoshal

The Problem

• Given a graph G(V,E) (directed or undirected), two nodes s,t ϵ V and a non-negative budget B, find an s-t walk of total length at most B so as to maximize the number of distinct nodes visited by the walk.

• A node may be visited multiple times by the walk but is only counted once in the objective function.

• Motivated from real world problems in vehicle routing, robot motion planning.

Quick Facts

• Is NP-hard

• Is APX-hard (cannot be approximated within 1481/1480)

• Introduced in 1987 by Bruce L. Golden, Larry Levy, and Rakesh Vohra

Introduction

• Orienteering belongs to the class of prize collecting TSP.

• Given a set of cities with some “prize” associated with each city and given a set of pair wise distances, a salesman needs to pick a subset of cities so as to minimize distance and maximize total reward.

• Bi-criteria optimization problem.

Introduction

• General approach:

• Fix one criteria and optimize the other– K-TSP, k-Stroll

Fix: No of nodes (total reward)

Optimize: distance

– OrienteeringFix: Total distance (budget)

Optimize: Reward (no of nodes covered)

The Story So far

• First non-trivial approximation: 2 + Ɛ (Arkin, Mitchell and Narasimhan) for points on Euclidean plane.

• 4 (Blum et al) for points on arbitrary metric spaces.

• 3 (Bansal et al)

• PTAS (K. Chen and Har-peled) for fixed dimensional Euclidean space.

Results

• Undirected graphs:– Ratio of (2+δ) and running time of nO(1/ δ^2)

• Directed Graphs:• Ratio O(log2 OPT)

Approach

• The basic approach:

Approximation of k-stroll -> approximation of minimum excess -> approximation for Orienteering.

The MIN_EXCESS Problem

• The excess of a path P is defined as the difference between the length of the path L and the shortest distance D between its end points. i.e excess(P) = L – D

• Given a weighted graph with rewards, end points s and t, and a reward quota k, find a minimum excess path from s to t collecting reward at least k.

MIN_EXCESS (Contd)

• If x is the excess of an optimal path, an α-approximation for the minimum-excess problem has length at most:

d(t) + αx ≤ α(d(t) + x)

and hence gives an α approximation for the minimum length problem.

Note: d(t) is the shortest distance between the end points of the path.

K-stroll to Orienteering via min-excess

1. In undirected graphs a β-approximation to the k-stroll problem implies a (3 β/2 – ½)-approximation to the minimum excess problem (Blum 2003)

2. In directed graphs a β-approximation to the k-stroll problem implies a (2β - 1)-approximation to the minimum excess problem.

3. A γ-approximation to the min-excess problem implies a | γ| approximation for orienteering. (Bansal 2004)

• 2 and 3 can be extended to show that an (α,β)-approximation to the k-stroll algorithm for directed graphs gives (α|2β -1|)-approximation for directed orienteering.

• Using 1 and 3 and a (1 + δ,2)-approximation for the k-stroll problem gives a ((1+ δ)*|2.5|) = (3 + δ)-approximation for orienteering.

• But we are interested in (2 + δ) approximation.

K-stroll to Orienteering via min-excess

(2 + δ) approach

• Begin with k-stroll

• Given a metric graph G, with 2 specified vertices s, t and a target k, find an s-t path of minimum length that visits at least k vertices.

• Let L be the length of such an optimal path and D be the shortest path distance from s to t.

(2 + δ) approach (Contd)

• Objective: For any fixed path that visits at least (1-O(δ))k vertices and has total length at most max1.5D,2L-D

• (Chaudhuri et al 2003) give a polynomial algorithm to find a tree T that spans k vertices containing both s and t, of length at most (1+ Ɛ)L for any Ɛ > 0

• Guesses O(1/Ɛ) vertices s,w1..wm,t such that an optimal path P visits the vertices in the given order and length and distance between any wi-wi+1 is < ƐL.

• Assume all edges in T are < ƐL.

• Let PTs,t be the path in T from s to t.

• L is the shortest s-t path visiting k vertices.

• Since length(T)<= (1+Ɛ)L

We can double all edges of T not on PTs,t to

obtain a path PT from s to t that visits k vertices.

Length of PT is 2length(T)-length(PTs,t) <=

2length(T) - D

Easy Doubling conditions

• If length(T) <= 5D/4 then PT has length at most 3D/2).

• Length(PTs,t) >= D+2ƐL then

length(PT) <= 2(1+Ɛ)L – (D + 2ƐL) = 2L - D

• If the easy doubling conditions are not met it means D <= 4/5(length(T)) and length(PT

s,t) >= (1/5 - 2Ɛ)L

• Modify the tree T to T’ in the following way:

• Greedily decompose the edge set of T\PTs,t into

Ω(1/δ) disjoint connected components, each with length in [δL,3δL)

• Merge connected components to get T’• Tree T’ contains a vertex of degree 1 or 2 that

corresponds to a component containing at most 32δk vertices.

• Remove C• C can either be a leaf or a node with degree 2

and contain at most 32δk vertices.• So we get a tree T’’ of length (1-32δ)k vertices.• If C is not a leaf we get two Trees.• Find the shortest distance between trees.• In either case double edges and we get a tree of

length at most 2L-D and we are done.

Conclusion

• Using results from Orienteering improved results can be obtained for TSP with deadlines and TSP with time windows.

References

• N. Bansal, A. Blum, S. Chawla and A. Meyerson. Approximation Algorithms for Deadline TSP and Vehicle Routing with time windows. Proc. Of ACM STOC 166-174. 2004

• A.Blum, S.Chawla, D. Karger, T.Lane, A. Meyerson and M. Minkoff. Approximation algorithms for Orienteering and disounted reward TSP, SIAM J. On Computing, 37(2):653-670,2007.

• K. Chaudhuri, B.Godfrey, S.Rao and K.Talwar. Paths, trees and minimum latency tours. Proc. of IEEE FOCS, 36-45,2003.