Approximation Algorithms for Path-Planning Problems with Nikhil Bansal, Avrim Blum and Adam Meyerson Shuchi Chawla Carnegie Mellon University

Approximation Algorithms for

Path-Planning Problems

with

Nikhil Bansal, Avrim Blum and Adam Meyerson

Shuchi Chawla

Carnegie Mellon University

Shuchi Chawla, Carnegie Mellon University2

The Trick-o-Treaters Problem

Collect as much candy as possible within 6pm and 8pm

More candy more popularity with the kids

Have limited time to look for the lost wallet different places have different likelihoods of containing it

Some complicating constraints Limited amount of time Cannot necessarily visit all locations

Path-planning: Given graph (metric) G, construct a path satisfying

some constraints and optimizing some function

The Lost-Wallet Problem


Path-planning in the real world

A robot-navigation problem Deliver packages to certain locations Faster delivery => greater happiness Limited battery power Packages have different deadlines for delivery

Assembly analysis

Manufacturing

Production planning


A reward-time trade-off

Given graph (metric) G, construct a “short” path that visits “many” nodes

Classic formulation – Traveling SalesmanFind the shortest tour covering all locations

Orienteering:Given a metric and a starting point, cover

as many “high-reward” locations as possible within a limited amount of time





Budget the path-length and maximize reward Orienteering Hard bound on path length

Time Window Visit node v within [Rv, Dv]

Impose a reward quota and minimize length

k-Path Collect at least k reward





Budget the path-length and maximize reward Orienteering 4 [Blum C Karger+03]

3 [Bansal Blum C Meyerson 04]

Time Window 3log2n [Bansal Blum C Meyerson 04]

Impose a reward quota and minimize length

k-Path 2 + [Chaudhury Godfrey Rao+ 03]


The rest of this talk

A 3-approximation for Orienteering

An O(log2n) approx for the Time-Window Problem

Orienteering with deadlines Incorporating release-dates

Extensions and Open Problems


Orienteering and k-Path

Orienteering : length · D ; maximize reward k-Path : reward ¸ k ; minimize length

Complementary problems

Series of results on k-TSP (related to k-Path)

[BRV99] [Garg99] [AK00] [CGRT03] …

best approx: (2+)

None for Orienteering until recently!


Why is Orienteering difficult?

First attempt – Use distance-based approximations to approximate reward

Let OPT(d) = max achievable reward with length d

A 2-approx for distance implies that ALG(d) ≥ OPT(d/2)

However, we may have OPT(d/2) << OPT(d) Bad trade-off between distance and reward!

sOPT(d)

APPROX



First attempt – Use distance-based approximations to approximate reward

Idea – Modify the algorithm itself Doesn’t help – moat-growing always goes for

shallow fruit

Orienteering is inherently harder; Perturbation of the input changes the output widely

sOPT(d)

APPROX



Second attempt – approximate subparts of the optimal path and shortcut other parts

If we stray away from the optimal path by a lot, we may not be able to cover reward that’s far away

Approximate the “extra” length taken by a path over the shortest path length

s tOPTAPPROX



Second attempt – approximate subparts of the optimal path and shortcut other parts

If we stray away from the optimal path by a lot, we may not be able to cover reward that’s far away

Approximate the “extra” length taken by a path over the shortest path length

If OPT obtains k reward with length d+, ALG should obtain the same reward with length d+

Min-Excess Path Problem


The Min-Excess Problem

Given graph G, start and end nodes s, t, reward on nodes v, target reward k, find a path that collects reward at least k and minimizes (P) = ℓ(P) – d(s,t)

At optimality, this is exactly the same as the k-path objective of minimizing ℓ(P)

However, approximation is different: Min-excess is strictly harder than K-path

There is a (2+)-approximation for Min-Excess

[Blum, C, Karger, Meyerson, Minkoff, Lane, FOCS’03]

Our algorithm returns a path with length

d(s,t) + (2+) (P)

excess


A 3-approximation to Orienteering

Construct a path from s to t, that has length D collects maximum reward

Given a 3-approximation to min-excess:1. Divide into 3 “equal-reward” parts (hypothetically)

2. Approximate the part with the smallest excess 3-approximation to orienteering

s t

Excess of one subpath · (1+2+3)/3Can afford an excess up to D – ℓwhite = 1+2+3

1 2

3

Excess of path P (P) = dP(u,v)– d(u,v)

Using an r-approx for Min-excess ( r Z+ ), we get an r-approximation for s-t Orienteering

v1

v2 OPT

APPROX

Open: Given an r-approx for min-excess (r 2 R +), can we get r-approx to Orienteering?


So far…

A 3-approximation for Orienteering You

An O(log2n) approx for the Time-Window Problem

Orienteering with deadlines Incorporating release-dates

Extensions and Open Problems

Coming up…

learnt how to look for your lost walletshould’ve


The Time-Window Problem

Find a path visiting many nodes in their time-window

school bus routing FEDEX dial-a-ride service newspaper delivery

Widely studied in scheduling and OR literature

Constant-approx known for points on a line, few different time-windows; No approximation known for the general case

A special case – The Deadline-TSP Problem Vertices only have deadlines All “release-times” are 0.


The next step: Deadline-TSP

Every vertex has a deadline D(v); Find a path that maximizes nodes v visited before D(v)

If the last node on the path has the min deadline, use Orienteering to approximate the reward

Everything visited before the minimum deadline Don’t need to bother about deadlines of other nodes

Does OPT always have a large subpath with the above property?

There are many subpaths of OPT with the above property that together contain all the reward

NO!


A segmentation of OPT

Time

Dead

line


Deadline-TSP

Segment graph into many parts, approximate each using Orienteering and patch them together

How do we find such a segmentation without knowing the optimal path?

In order to avoid double-counting of reward, segments should be node-disjoint

Our result – There exists a segmentation based only on deadlines, such that the resulting solution is a (3 log n)-approximation


A 2-dimensional view

Time

Dead

line

minimal vertices

“Disjoint Rectangles”


The Rectangle Argument

Approximate reward contained in a family of disjoint rectangles Every pair of rectangles is non-overlapping in

BOTH dimensions

We construct O(log n) families of disjoint rectangles1. These cover ALL the reward in OPT2. We can approximate the best of them

We get an O(log n)-approximation



1. There are O(log n) families of disjoint rectangles that cover all the reward in OPT

Time

Dead

line



1. There are O(log n) families of disjoint rectangles that cover all the reward in OPT

Time

Dead

line

If there are between 2b and 2b+1 points in between, then either the bth or a larger family contains exactly 1 point in the interval



2. We can approximate the best disjoint family

Suppose we know the minimal vertices Just try out all the log n families

Problem - Minimal vertices depend on the optimal tour!

Solution – Try all possibilities. They are ordered by deadlines, so use a simple dynamic program



2. We can approximate the best disjoint family

Time

Dead

line


The O(log n)-approximation

Approximate reward contained in a “disjoint” family of rectangles Every pair of rectangles is non-overlapping in

BOTH dimensions

We construct O(log n) families of disjoint rectangles1. These cover ALL the reward in OPT2. We can approximate the best of them

Obtain an O(log n)-approximation


From Deadlines to Time-Windows

Nodes have deadlines as well as release times

Release times are dual to deadlines – if we look at the path from the end to the start, release times become deadlines!

Log-approximation for deadlines log-approximation for release dates

Algorithm for Time-Windows: Run the approximation for Deadline-TSP Replace Orienteering by Orienteering with release-dates

O(log2n)-approximation for the Time-Window problem

s tOPT

ℓ(OPT) = L

v

Require ℓ(s,v) R(v) ℓ(t,v) L-R(v)

D(v) = L-R(v)

st


A Bicriteria Approximation

Given any > 0,

Get O(log 1/) fraction of reward

Exceed deadlines by a (1+) factor

O( log Dmax )-approximation

Constant factor approximation if we can exceed deadlines by a small constant factor

Nice trade-off:Halving the extra time taken, increases

the approximation factor by only an additive 1


Deadline TSP 3 log n

An overview of our results

Time-Window Problem 3 log2n

ApproximationProblem

Orienteering 3

Time-Window Problem - bicriteria

reward: log 1/ deadlines: 1+


Future Directions

Better approximations constant factor for Time-Windows? special metrics such as trees or planar graphs

Hardness of approximation log-hardness for Time-Windows?

Asymmetric Path-planning the graph is directed; still obeys triangle inequality


Questions?

Documents

Approximation Algorithms for Path-Planning Problems with Nikhil Bansal, Avrim Blum and Adam Meyerson Shuchi Chawla Carnegie Mellon University