Minimum Partition Partitions of s of Minimum Integer Sequences - - PowerPoint PPT Presentation

Minimum Minimum Partition Partitions of s of Integer Integer Sequences Sequences

Ronny Hansmann & Uwe Zimmermann

Institute for Mathematical Optimization TU Braunschweig

Basic Problem Basic Problem

The underlying problem: Given: A sequence of integers Goal: Find a Minimum Partition of the sequence into subsequences satisfying certain feasibility requirements Crucial parameters: Integers are distinct or not distinct? Feasibility requirements

Previous Previous Works Works

For sequences of distinct integers, i.e., permutations: Papers on Permutations:

Erdös/Szekeres 1935, Even et al 1971, Chung 1980, …

Papers on Stack Sorting:

Tarjan 1972, Avis/Newborn 1981, Atkinson 2004, …

Papers on Shunting:

Winter/Zi 2000, Di Stefano/Koci 2003, …

To be published in proceedings of LATIN 2006:

Di Stefano, Krause, Lübbecke, Zi

Theoretical Theoretical Results Results for for Permutations Permutations

Di Stefano, Krause, Lübbecke, Zi, 2005 k-modal subsequences Di Stefano, Zi, 2004 Unimodal subsequences Wagner, 1984 Monotone subsequences Even et al, 1971 Isotone subsequences Complexity Feasibility requirement

O(n log n) N P-complete

strongly

N P-complete

strongly

N P-complete

strongly

Practical Practical Results Results for for Permutations Permutations

Exact: MIP-Modells Set Covering Minimum Graph/Hypergraph-Colorings Minimum Cost Flow with GUB‘s Very Bad Bad Effective Approximations LP-Rounding Monotone subsequences: factor 2 k-modal subsequences: factor Fomin et al 2002 Monotone subsequences: factor 1,71 Fast Fast

...

2(k + 1)

Practical Practical Results Results for for Permutations Permutations

Bad Online-Algorithms No constant factor competitive online algorithms for the hard problems Next-Fit & Best-Fit: Isotone subsequences: optimal Monotone case: -competitive (tight) Very Good Heuristics Greedy

n 4

Not Not-

• distinct

distinct Integer Integer Sequences Sequences

New: Sequences of not necessarily distinct integers What is our motivation?

Practical Practical Background Background

Relate to shunting problems of our practical project (together with BASF, Ludwigshafen): A sequence of waggons (not distinct integers) has to be stored on a minimal number of tracks in a shunting yard such they can depart as desired without shunting

1 3 2 1

T Train rain S Sort

• rt S

Shunting hunting P Problem roblem

Basic feasibility requirement: Only complete and sorted trains (waggons of same integer value) are pulled out

1 1 1 1

Basic Basic Notations Notations

• … sequence of n integers

(waggons)

S = (s1,.. .,sn)

• with

is called subsequence of S

S ⊇ (si1, si2, . . . , sin)

i1 < i2 < · · · < in

• … set of t different integers

(trains)

si ∈ T = {1,. ..,t}

Parameters Parameters

Does the order of the outgoing trains matter? no: Non-chronological yes: Chronological Is there a separation between arrival and departure? yes: Sequential no: Concurrent

⇒ Five versions

N Non

• n-
• chronological

chronological & & S Sequential equential ( (NS NS) )

1 2 4 3 1 2 3 2 1 4 3

Non-chronological: Order of outgoing trains does not matter Sequential: First departure after last arrival

1 1 4 4 1 4 4

Trains u and v can not be placed on the same track

⇔

(u,v,u) : u 6= v is a subsequence of S

N Non

• n-
• chronological

chronological & & C Concurrent

• ncurrent (

(NC NC) )

1 2 3 2 1 4 3

Non-chronoligal: Order of outgoing trains does not matter Concurrent: No separation between arrival and departure

1 4 1 4 4 2 1 4 3 1 4

Trains u and v can not be placed on the same track

⇔

is a subsequence of S (u,v,u,v) : u 6= v

Chronological Chronological & & S Sequential equential ( (S S) )

1 3 1 2 3 2 1 4 3

Chronological: Order of outgoing trains matters Sequential: First departure after last arrival

1 1 4 1 4 4 2 4 4

Trains u and v can not be placed on the same track

⇔

is a subsequence of S (u,v) : u < v

Chronological Chronological & & C Concurrent

• ncurrent (

(C C) )

1 2 4 3 1 2 3 2 1 4 3

Chronological: Order of outgoing trains matters Concurrent: No separation between arrival and departure

1 1 4 4 1 4 4

Trains u and v can not be placed on the same track

⇔

is a subsequence of S (u,v,w) : w · u < v

T Time ime W Windows ( indows (TW TW) )

1 2 4 3 1 2 3 2 1 4 3 1 1 4 4 1 4 4 1 2 4 1 3 4 2 1 4 3 1 4

time

„forbidden“

Minimum Minimum Partitions Partitions

The versions NS, S, NC, and C are equivalent to: Find a Minimum Partition of the given sequence into subsequences such that: All elements with identical value belong to the same subsequence If two elements and start a forbidden subsequence

• f S, they

do not both belong to the same subsequence

si = u sj = v (u,v,. ..)

Minimum Minimum Partitions Partitions

The version TW is equivalent to: Find a Minimum Partition of the given set of intervals into subsets such that: All intervals with identical end point belong to the same subset If two intervals overlap, they do not both belong to the same subset

Graph Graph Coloring Coloring Formulation Formulation

For our problems we define a graph with vertices corresponding to the trains We add an edge if there exists a forbidden subsequence:

C S NC NS u=si, v=sj : Ii and Ij overlap TW

• r if:

Min Colorings of these graphs are optimal solutions for our problems

(u,v)

(u,v,u) : u 6= v (u,v,u,v) : u 6= v (u,v) : u < v (u,v,w) : w · u < v

Time Windows vs. Time Windows vs. Sequences Sequences

Obvious Obvious Relations Relations

NS TW NC S C

z∗

NS(SI) ≥ z∗ NC(SI)

z ∗

S(SI) ≥ z ∗ T W (I) ≥ z∗ C(SI)

1 4 4 1 2 3 1 1 2 3 4 4 1 4 4 1 2 3 1 4 4 1 2 3 1 4 4 1 2 3 1 1 2 3 4 4 1 4 4 1 2 3 1 1 3 2 4 4 1 4 4 1 2 3 1 4 4 1 2 3

z∗

NS(SI) · z∗ S(SI)

z∗

NC(SI) · z∗ C(SI)

C: NC: TW:

Complexity Complexity

NS:

Reduction to Coloring of Overlap Graphs Reduction to Coloring of Overlap Graphs

S:

Graphs are perfect BEST FIT is optimal

Interval Coloring of

Intervals given sorted C is a special case

N P-complete

P

N P-complete N P-complete O(n2) O(n log n)

Exact Exact Solution Solution Methods Methods

For NS, S: FIRST FIT and BEST FIT For the

• complete versions NC, C, TW:

Graph Coloring (weak formulation) Minimum Cost Flow with side constraints

N P

Minimum Minimum Cost Cost Flow Flow (NC) (NC)

q s

3, 5, 2, 5, 4, 2, 4, 2, 5, 1, 3, 1,

S = ( ) =

· 1 · 1

P

3

= 1 P

2

= 1 P

1

= 1 P

5

= 1 P

4

= 1

2, 2, 5, 5,

forbidden: (u,v, u,v) : u 6= v

Minimum Minimum Cost Cost Flow Flow

This Formulation based on Minimum Cost Flow with side constraints can be applied to NC, C, and TW These Minimum Cost Flow Problems with side constraints solve the respective versions NC, C, and TW optimally.

Computational Computational Results Results

NS NC

# waggons tracks 20 1 – 5 2 – 12 3 – 28 100 300

FIRST FIT

< 1 ms

Network-MIP Color-MIP FIRST FIT

< 1 ms # trains tracks time [sec] time [sec] tracks 10 1 – 5 0,01 0,01 – 0,11 100%: = 30 2 – 19 0,01 – 0,02 0,01 – 1,01 50%: = 100%: <+1 100 3 – 45 0,13 – 3,05 0,02 – > 2h 50%: <+1 100%: <+4

Computational Computational Results Results

Color-MIP # trains 10 30 100 # waggons 20 100 300

S TW C

BESTFIT < 1 ms Network-MIP FIRSTFIT < 1 ms Network-MIP Color-MIP FIRSTFIT < 1 ms tracks tracks time [sec] time [sec] track tracks time [sec] time [sec] tracks 7 – 10 1 – 8 0,01 0,01 100%: = 1 – 5 0,01 0,01 100%: = 25 – 30 2 – 26 0,01 0,01 – 1,09 90%: = 100%: <+1 2 – 15 0,01 – 0,04 0,01 – 0,51 90%: = 100%: <+1 72 – 100 4 – 74 0,04 - 1,04 0,04 - >2h 50%: = 100%: <+3 3 - 35 0,14 - 3,69 0,03 - >2h 50%: <+1 100%: <+3

Online Online-

• Algorithms

Algorithms

We say an Online-Algorithm is sensible if it never yields a worst case solution, that is, one with maximal possible gap to optimality In our case: Optimum: 1 Track, Online-Solution: T Tracks Strict Online-Interpretation: No information of S in advance No sensible Online-Algorithms for our problems.

Online Online-

• Algorithms

Algorithms

Soft Online-Interpretation: Last incoming waggon

• f a train is signed

FIRST FIT is optimal for online version of NS and BEST FIT for online version of S No sensible Online-Algorithms for NC,C, and TW

Conclusion Conclusion & Outlook & Outlook

Conclusion: We can solve all our versions with practically relevant input size within a few seconds Outlook: Extend problems to more complex ones of our practical project Open theoretical questions: Other topologies: queues,… Bounded case Multiple sortings