Modelli e Metodi di Ottimizzazione per la Biologia Computazionale e

la Medicina

Giuseppe LanciaUniversità di Udine

MOLECULAR BIOLOGY

• Human Genome Project (1990): read and understand human (and other species) genome (DNA)

• Scientific and practical applications (medicine, agriculture, forensic, ….)

multi-million project involving several countries

COMPUTER SCIENCE

To meet the goal we need computers and programs to deal with problems such as

- Huge data sets (billions of informations to be analized)

- Data Interpretation (contradictory, erroneous or inconsistent data)

- Data sharing and networks (online genomic data banks)

Computational (molecular) Biology

• Combinatorics, Discrete Math• Combinatorial Optimization• Integer Programming• Complexity theory (Approximations and Hardness)• Graph theory• (but also Stringology, Data Bases, Neural Networks....)

“Optimization problems arising in the analysis, interpretation and management of large sets of

genomic data”

FIRST PHASE COMPLETION : HUMAN GENOME SEQUENCE - 2001

World Consortium universities and labs

Celera Genomics (Craig Venter)

FIRST PHASE COMPLETION : HUMAN GENOME SEQUENCE - 2001

Computational Biology born around ’80-’90

• Algorithmic approaches (e.g. Dynamic Programming for alignment)

• Computational complexity (e.g. NP-hardness of folding)

• String-related problems, Information retrieval, Genomic data base….

• ……

mostly computer scientists dominated the field

• Some NP-hard problems in C.B. are OPTIMIZATION PROBLEMS

• These can be solved via mathematical programming techniques

• INTEGER LINEAR PROGRAMMING

• LAGRANGIAN RELAXATION

• SEMIDEFINITE PROGRAMMING

• QUADRATIC PROGRAMMING

O.R. people (and O.R. techniques) entered the field

The most important approach of this type is probably Integer Linear Programming

It allows the solution of NP-hard problems via specialized Branch and Bound algorithms

The lower bound comes from the Linear Programming relaxation of the model (and can be computed in polynomial time)

The I.P. approach

1. Define integer (usually binary) variables

The I.P. approach

1. Define integer (usually binary) variables

The I.P. approach

2. Define linear constraints that feasible solutions must satisfy

1. Define integer (usually binary) variables

The I.P. approach

2. Define linear constraints that feasible solutions must satisfy

3. Define linear objective function that optimal solution must mini(maxi)mize

1. Define integer (usually binary) variables

The I.P. approach

2. Define linear constraints that feasible solutions must satisfy

3. Define linear objective function that optimal solution must mini(maxi)mize

4. Solve by Branch and Bound. The LP relaxation (remove integrality requirements on variables) gives the bound

1. Define integer (usually binary) variables

The I.P. approach

2. Define linear constraints that feasible solutions must satisfy

3. Define linear objective function that optimal solution must mini(maxi)mize

4. Solve by Branch and Bound. The LP relaxation (remove integrality requirements on variables) gives the bound

There can be an exponential number of variables. We need to make them implicit BRANCH-AND-PRICE

1. Define integer (usually binary) variables

The I.P. approach

2. Define linear constraints that feasible solutions must satisfy

3. Define linear objective function that optimal solution must mini(maxi)mize

4. Solve by Branch and Bound. The LP relaxation (remove integrality requirements on variables) gives the bound

There can be an exponential number of variables. We need to make them implicit BRANCH-AND-PRICE

There can be an exponential number of constraints. We need to make them implicit BRANCH-AND-CUT

Some Integer Programming models in C.B.Haplotyping Clark’s rule (Gusfield ) Parsimony (Gusfield, Lancia+Pinotti+Rizzi, Brown+Harrower) Fragment assembly (Lancia+Bafna+Schwartz+Istrail)

Protein folding Energy potentials (Wagner+Meller+Elber) Folding ab initio (Carr+Hart+Greenberg) Threading (RAPTOR, Xu+Li+Kim+Xu) Docking (Doye+Leari+Locatelli+Schoen, Althaus+Kohlbacher+Lenhof+Muller ) Fold comparison (Carr+Lancia+Istrail+Walenz,Caprara+Lancia) Sequence Alignment and consensus Lenhof+Vingron+Reinert Althaus+Caprara+Lenhof+Reinert Fischetti+Lancia+Serafini Meneses+Lu+Oliveira+Pardalos

Physical Mapping Alizadeh+Karp+Weisser+Zweig

Genome Rearrangements Caprara+Lancia

Haplotyping Clark’s rule (Gusfield ) Parsimony (Gusfield, Lancia+Pinotti+Rizzi, Brown+Harrower) Fragment assembly (Lancia+Bafna+Schwartz+Istrail)

Protein folding Energy potentials (Wagner+Meller+Elber) Folding ab initio (Carr+Hart+Greenberg) Threading (RAPTOR, Xu+Li+Kim+Xu) Docking (Doye+Leari+Locatelli+Schoen, Althaus+Kohlbacher+Lenhof+Muller ) Fold comparison (Carr+Lancia+Istrail+Walenz,Caprara+Lancia) Sequence Alignment and consensus Lenhof+Vingron+Reinert Althaus+Caprara+Lenhof+Reinert Fischetti+Lancia+Serafini Meneses+Lu+Oliveira+Pardalos

Physical Mapping Alizadeh+Karp+Weisser+Zweig

Genome Rearrangements Caprara+Lancia

We’ll see some examples

BIOLOGY 101

A genome is a long string over the DNA alphabet {A,C,G,T}

In man it is some 3.000.000.000 letters

DNA is responsible forour diversity as well asour similarity

Small changes in a genomecan make a big difference,like from... to...

CELL

Nucleus

Chromosomes (pairs)

TCATCGAAGTAGCT

Eukariotic diploid organisms

THE CENTRAN DOGMA OF MOLECULAR BIOLOGY

attagcatggatagccgtatatcgttgatgctggataggtatatgctagatcgatggcaatta intronsexons

attag|ctatatcgttgatg|tatatgcta|cga|aatta A GENE

THE CENTRAN DOGMA OF MOLECULAR BIOLOGY

attagcatggatagccgtatatcgttgatgctggataggtatatgctagatcgatggcaatta intronsexons

attag|ctatatcgttgatg|tatatgcta|cga|aatta

att agc tat atc gtt gat gta tat gct acg aaa tta codon triplets

R N C A S S F C W Y Q V

A GENE

amino acids:a PROTEIN

THE CENTRAN DOGMA OF MOLECULAR BIOLOGY

attagcatggatagccgtatatcgttgatgctggataggtatatgctagatcgatggcaatta intronsexons

attag|ctatatcgttgatg|tatatgcta|cga|aatta

att agc tat atc gtt gat gta tat gct acg aaa tta codon triplets

R N C A S S F C W Y Q V

A GENE

amino acids:a PROTEIN

The protein folds to a 3D shape to perform its function

CENTRAL DOGMA:

1 gene 1 protein

From DNA to strings (i.e. reading our genome): Shotgun sequencing

From DNA to strings (i.e. reading our genome): Shotgun sequencing

amplification

fragmentation

From DNA to strings (i.e. reading our genome): Shotgun sequencing

amplification

TGAGCCTAG GATTT GCCTAG CTATCTTATAGATA GAGATTTCTAGAAATC ACTGA TAGAGATTTC TCCTAAAGAT CGCATAGATA

fragmentation

sequencing

From DNA to strings (i.e. reading our genome): Shotgun sequencing

amplification

TGAGCCTAG GATTT GCCTAG CTATCTTATAGATA GAGATTTCTAGAAATC ACTGA TAGAGATTTC TCCTAAAGAT CGCATAGATA

fragmentation

sequencing

From DNA to strings (i.e. reading our genome): Shotgun sequencing

amplification

assembly

We need to merge the fragments in order to retrieve the original DNA sequence

-50,000,000 fragments-1000 char each…

We better use computers!

assemblyACTGCATTAGCGAGTTATAGATCGAGTAGAGATATCGCGGGGACTGCATTAGCGAGTTATAGATCGAGTAGAGATATCGCGGGGACTGCATTAGCGAGTTATAGATCGAGTAGAGATATCGCGGGGACTGCATTAGCGAGTTATAGATCGAGTAGAGATATCGCGGGG

- Take 10 copies of «Corriere della Sera»- Cut each in tiny pieces (1cm2)- Put the pieces in a bag and shuffle

- Grab five handful of pieces and throw them awayProblem: retrieve the newspaper from the remaining pieces

• Let’s make the problem harder:- Ultra-tiny pieces (1mm2)

Difficulties : Ripeated words (e.g., «ha», «dopo», «quando», «governo»…)

- It is not the CDS, but the encyclopedia Treccani (20 volumes)- It is written in chinese!!

Still thr problem would be easier than sequencing the humangenome

Understanding the problem

Assembly

• Repeated words and missing words create problems to be solved by sophisticated programs, based on statistics and mathematical models.

• The basic underlying problem (notwithstanding the above complications) is called Shortest Superstring Problem (SSP)

Shortest superstring:

• Given a set of strings s1, .., sn, find a string s that contains each si as a substring..

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

Shortest superstring:

cane nerone onere rene neon

reneroneoncaneonere -> 19

caneronereneonere ->17

caneroneonerene -> 15

acct cattgt gtgcca cctg cattgtgccacctg

The space of potential (non-redundant) solutions has size O(n!)

The problem is NP-Hard

There is an effective greedy algorithm

Shortest superstring:

cane nerone onere rene neon

Greedy algorithm:- Merge the two strings with max overlap. Replace them with their fusion. - Repeat until only one string is left.

cane nerone onere rene neon

neronere

Greedy algorithm:- Merge the two strings with max overlap. Replace them with their fusion. - Repeat until only one string is left.

cane nerone onere rene neon

neronere

neronerene

Greedy algorithm:- Merge the two strings with max overlap. Replace them with their fusion. - Repeat until only one string is left.

cane nerone onere rene neon

neronere

neronerene

neronereneon

Greedy algorithm:- Merge the two strings with max overlap. Replace them with their fusion. - Repeat until only one string is left.

cane nerone onere rene neon

neronere

neronerene

neronereneon

caneronereneon

Greedy algorithm:- Merge the two strings with max overlap. Replace them with their fusion. - Repeat until only one string is left.

• In this case it found the optimum, but there is no guarantee• There’s, however, a guarantee that v(GREEDY) <= 2.5v(OPT) (i.e., it is a 2.5-APPROX ALGORITHM (Sweedyk))

• OPEN PROBLEM (Conjecture holding from > 20 years):

Prove that GREEDY is a 2-APPROX ALGORITHM

For more info seehttp://fileadmin.cs.lth.se/cs/Personal/Andrzej_Lingas/superstring.pdf

Greedy algorithm:

Sequence Alignments

• Sequences evolve and change• E.g.: deletion, insertion, mutation

attcgattgat attcggat deletion

attcgattgat attcggat insertion

attcg atgcg mutation

Given two genomic sequences (e..g., man and mouse)we want to compare them

Languages evolve as well

• Mater• Madre• Mother• Matre• Mutter

• Mater• Madre• Mother• Matre• Mutter

Good alignment:

M A T - E RM O T H E R

Languages evolve as well

• Mater• Madre• Mother• Matre• Mutter

Good alignment:

M A T - E RM O T H E R

Bad alignment:

- - M A T - E RM O T H - E R -

Languages evolve as well

• Mater• Madre• Mother• Matre• Mutter

Good alignment:

M A T - E RM O T H E R

Bad alignment:

- - M A T - E RM O T H - E R -

Multiple alignment:

M A T - E R -M O T H E R -M A D - - R EM A T - - R EM U T T E R -

Languages evolve as well

• Mater• Madre• Mother• Matre• Mutter

Good alignment:

M A T - E RM O T H E R

Bad alignment:

- - M A T - E RM O T H - E R -

Multiple alignment:

M A T - E R -M O T H E R -M A D - - R EM A T - - R EM U T T E R -

If DNA sequences:

A C T - G G -A C T C G G -A G T - - C TC C T - - G TA - T T C G -

Languages evolve as well

• Mater• Madre• Mother• Matre• Mutter

Good alignment:

M A T - E RM O T H E R

Bad alignment:

- - M A T - E RM O T H - E R -

To see if an alignment is good: score 0 identical(or similar) letters (e.g. T-D, B-P, S-F) and score 1different or deleted letters.

Cost: 1 Cost: 8

Languages evolve as well

• Mater• Madre• Mother• Matre• Mutter

Good alignment:

M A T - E RM O T H E R

Bad alignment:

- - M A T - E RM O T H - E R -

To see if an alignment is good: score 0 identical(or similar) letters (e.g. T-D, B-P, S-F) and score 1different or deleted letters.

Cost: 1 Cost: 8

There is an exponential number of possible alignments of two strings, but finding the optimal one is easy (i.e. polynomial). We’ll see it next.For many strings, there is no polynomial algorithm (if P =/= NP), but there are effective heuristics andapproximation algorithms.

Languages evolve as well

The basic problem: pairwise sequence alignment:

Seq. 1: ATCGGCTTGTTATTG

Seq. 2: ATGGGATTTAATTGCCC

ATCGGCTTGTTA-TTG--- ATGGGAT--TTAATTGCCC

Alignment:

The basic problem: pairwise sequence alignment:

Seq. 1: ATCGGCTTGTTATTG

Seq. 2: ATGGGATTTAATTGCCC

ATCGGCTTGTTA-TTG--- ATGGGAT--TTAATTGCCC

Alignment:

Cost c(x,y) for each pair x, y in {A, T, C, G, -}

Problem: find the arrangement minimizing Si c(AL[1,i], AL[2,i])

This value is the edit distance d(S’, S”)

Simple Dynamic Programming Recursion:

d(S’[1…n], S”[1..m]) = min {d(S’[1..n-1],S”[1…m]) + c(S’[n],-)

d(S’[1..n],S”[1…m-1]) + c(-, S”[m])

d(S’[1..n-1],S”[1…m-1]) + c(S’[n],S”[m])

d(ATCCG, ATTC) = min {d(ATCC, ATTC) + c(G,-)

d(ATCCG, ATT) + c(-, C)

d(ATCC, ATT) + c(G,C)

Cost: O(n m) (Smith & Waterman, ’86)

This can be extended to Multiple Sequence Alignment

Useful for:

• Finding conserved patterns (functionally relevant)• Clustering genomic sequences (e.g. protein families)• Evolutionary studies (e.g. intra-species comparisons)• ….

ATCCGTTAGTTA---GATTG---ATT--TTAGTTA---GATTGCACATTCGTTAGTTAGTAGATTGCA-TTTCGTTA--TA-TACATTGCAC

This can be extended to Multiple Sequence Alignment

Useful for:

• Finding conserved patterns (functionally relevant)• Clustering genomic sequences (e.g. protein families)• Evolutionary studies (e.g. intra-species comparisons)• ….

Unfortunately…it’s NP-hard (Wang & Jiang, ’94) (Dynamic Programming costs O(2 n ) for k sequences of length n) k k

ATCCGTTAGTTA---GATTG---ATT--TTAGTTA---GATTGCACATTCGTTAGTTAGTAGATTGCA-TTTCGTTA--TA-TACATTGCAC

The distance in the alignment A is denoted dA(S’,S”). We wish tominimize SP(A) = S{S’,S”} dA(S’,S”) (sum-of-pairs objective)

a:humanafrican cox1 ;----accatccatcaatctgcgatccccctccaataccaaccataccactcctggccttccagttb:humprobeuro cox1 ;gtacaccatccatcaa--tgcgatccccctccaataccaaccataccactcctggccttccagttc:gorilla cox1 ;acatattacccatcaacttataatctctttccagcatcagtcatgctattcccgaccccttagttd:gibbon cox1 ;gcgtattactcgctggcccatacctcttcctagaccccaactgtgtcgcctctgattccctagtte:chimpanzee cox1 ;acacactacctatcgacttacaattccctcccaacaccagtcatgctgttcccgacccctcaactf:pigmychimp cox1 ;---cactacccattgacttacaatcccttctcagcactaatcacgctgttcccgatccctcaactg:baboon cox1 ;atgcaccgtttactagctcataccttccccta--ccctgatcgcatcatctttaatcccctaacch:orangutan cox1 ;gcgcgctgtccatcatcccacacttctctcccaacattggctataccgtccttgactccctcatci:sumorang cox1 ;gcgcgccgtccgtcatcccacacctctctcccaacatcaactataccacccctaactccctcgtc

agccccccccctagctgcctcatctccacccgcaccctcttctcccccccccaaacctcctctttac---agccccccccctagctgcctcatctccacccgcaccctcttctcccccccccaaacctcctctttactttaatcctccttccaatcgccctacctccacccgcactttctttctctccccccaaattcctttctcaccttgaccctcctttcggccgttctcccccctctcgcatcccaccttcccccttttaaactctccccttactccAacgtcccccctgacca--ccattttcacccgcactcccttcctttcctcccgaacttttttcttatcttaatgtcccccctggccactccactttcacccgcattcccttcctttcctcctgaacttttttctcatcttggtccccccctcaaccaccctcccccatttaatatctcaccctctcctttccaagcttccccctcatcccAactcttacctcaatcgtctta— tca-tcaacgcccccctcccccattattatgttccttctcccccccaactcttaccttaaccatctcatttcatccagtgcccccctccctcaccatcatatcctctttccccccc

Example: Same protein, different species

Sometimes the sequences can be related via a Phylogenetic Tree

A tree can be used to align sequences. Here is how:

Theorem 1: given a fully labeled tree, there is a multiple alignmentA(T) such that dA(T) (S(i),S(j)) = d(S(i), S(j)) for all edges (i,j) of T

Proof (by 1 drawing):

ATGGC TTGGAC

ATG-GCTTGGAC--

-TTGGACAT-GG-C

-TTG-GAC--AT-G-G-C--

A 2-approximation algorithm for multiple alignment and a goodheuristic borrow concepts from Network Design

3

2

4

10

5 92

7

6

3

The Routing Cost of a tree is the sum of distances in the tree between all pairs

XY

E.g. rc(X,Y) = 3 + 5 + 9 + 2 = 19. rc(T) = 1616

1

If we have k sequences, and a tree connecting them, we can align them using Theorem 1

tccat

attcgg

ttg

ttatcg

ttgaattcg

ttgattattc

ctt

tcct

ctggat

aatgt

The cost of edges {S’,S”} is the edit distance d(S’S”)

If we have k sequences, and a tree connecting them, we can align them using Theorem 1

tccat

attcgg

ttg

ttatcg

ttgaattcg

ttgattattc

ctt

tcct

ctggat

aatgt

By triangle inequality, dA(S’,S”) £ S {i,j}in path S’,..,S” dA(i,j)

dA(ttatcg, ctt) £ dA(ttatcg, ttga) + dA(ttga, attcg) + dA(attcg,ctt) = = d(ttatcg, ttga) + d(ttga, attcg) + d(attcg,ctt)

Hence SP(A(T)) £ rc(T)

Stars already yield a 2-approximation (next). If T* minimizes rc(T), we can expect a low SP(A(T*)). Finding T* is NP-hard, but “easier” than solving min SP

The cost of edges {S’,S”} is the edit distance d(S’S”)

Some particular trees (STARS) give a 2-approximation to SP

c

𝑅𝐶 (𝑆 (𝑐 ))¿=(𝑛−1)∑𝑖𝑑(𝑖 ,𝑐)

Some particular trees (STARS) give a 2-approximation to SP

c

𝑅𝐶 (𝑆 (𝑐 ))¿=(𝑛−1)∑𝑖𝑑(𝑖 ,𝑐)

)) =

Hence ,𝑅𝐶 (𝑆 (𝑐∗ )) ≤𝑎𝑣𝑔(𝑅𝐶)=1𝑛 ((𝑛−1)∑

𝑐∑𝑖𝑑(𝑖 ,𝑐))

Some particular trees (STARS) give a 2-approximation to SP

c

𝑅𝐶 (𝑆 (𝑐 ))¿=(𝑛−1)∑𝑖𝑑(𝑖 ,𝑐)

)) =

Hence ,𝑅𝐶 (𝑆 (𝑐∗ )) ≤𝑎𝑣𝑔(𝑅𝐶)=1𝑛 ((𝑛−1)∑

𝑐∑𝑖𝑑(𝑖 ,𝑐))

So, also SP

Some particular trees (STARS) give a 2-approximation to SP

c

𝑅𝐶 (𝑆 (𝑐 ))¿=(𝑛−1)∑𝑖𝑑(𝑖 ,𝑐)

)) =

Hence ,𝑅𝐶 (𝑆 (𝑐∗ )) ≤𝑎𝑣𝑔(𝑅𝐶)=1𝑛 ((𝑛−1)∑

𝑐∑𝑖𝑑(𝑖 ,𝑐))

So, also SP

But, for all A, SP

Some particular trees (STARS) give a 2-approximation to SP

c

𝑅𝐶 (𝑆 (𝑐 ))¿=(𝑛−1)∑𝑖𝑑(𝑖 ,𝑐)

)) =

Hence ,𝑅𝐶 (𝑆 (𝑐∗ )) ≤𝑎𝑣𝑔(𝑅𝐶)=1𝑛 ((𝑛−1)∑

𝑐∑𝑖𝑑(𝑖 ,𝑐))

So, also SP

But, for all A, SP

Some particular trees (STARS) give a 2-approximation to SP

c

𝑅𝐶 (𝑆 (𝑐 ))¿=(𝑛−1)∑𝑖𝑑(𝑖 ,𝑐)

)) =

Hence ,𝑅𝐶 (𝑆 (𝑐∗ )) ≤𝑎𝑣𝑔(𝑅𝐶)=1𝑛 ((𝑛−1)∑

𝑐∑𝑖𝑑(𝑖 ,𝑐))

So, also SP

But, for all A, SP

And so 2 OPT

Some particular trees (STARS) give a 2-approximation to SP

c

𝑅𝐶 (𝑆 (𝑐 ))¿=(𝑛−1)∑𝑖𝑑(𝑖 ,𝑐)

)) =

Hence ,𝑅𝐶 (𝑆 (𝑐∗ )) ≤𝑎𝑣𝑔(𝑅𝐶)=1𝑛 ((𝑛−1)∑

𝑐∑𝑖𝑑(𝑖 ,𝑐))

So, also SP

But, for all A, SP

And so 2 OPT

In conclusion, SP

ILP for Minimum Routing Cost Tree (Fischetti, L, Serafini)

Given G=(V,E), edge lengths d. Use 0-1 variables for each path 𝑝

ILP for Minimum Routing Cost Tree (Fischetti, L, Serafini)

Given G=(V,E), edge lengths d. Use 0-1 variables for each path

min ∑𝑝∈𝑃

𝑑𝑝 𝑥𝑝

𝑝

ILP for Minimum Routing Cost Tree (Fischetti, L, Serafini)

Given G=(V,E), edge lengths d. Use 0-1 variables for each path

(i)

min ∑𝑝∈𝑃

𝑑𝑝 𝑥𝑝

=1 for all 𝑖 , 𝑗∈𝑉

𝑝

ILP for Minimum Routing Cost Tree (Fischetti, L, Serafini)

Given G=(V,E), edge lengths d. Use 0-1 variables for each path

(i)

(ii)

min ∑𝑝∈𝑃

𝑑𝑝 𝑥𝑝

=1

∑𝑝∈𝑃 ( 𝑖 , 𝑗 ) :𝑒∈𝑝

𝑥𝑝 ≤𝑥𝑒

for all 𝑖 , 𝑗∈𝑉

e

𝑝

ILP for Minimum Routing Cost Tree (Fischetti, L, Serafini)

Given G=(V,E), edge lengths d. Use 0-1 variables for each path

(i)

(ii)

(iii)

min ∑𝑝∈𝑃

𝑑𝑝 𝑥𝑝

=1

∑𝑝∈𝑃 ( 𝑖 , 𝑗 ) :𝑒∈𝑝

𝑥𝑝 ≤𝑥𝑒

∑𝑒∈𝐸

𝑥𝑒=𝑛−1

for all 𝑖 , 𝑗∈𝑉

e

𝑝

ILP for Minimum Routing Cost Tree (Fischetti, L, Serafini)

Given G=(V,E), edge lengths d. Use 0-1 variables for each path

(i)

(ii)

(iii)

Pricing becomes O(n^2) shortest path problems with lengths de + ve(ve dual vars for (ii))

min ∑𝑝∈𝑃

𝑑𝑝 𝑥𝑝

=1

∑𝑝∈𝑃 ( 𝑖 , 𝑗 ) :𝑒∈𝑝

𝑥𝑝 ≤𝑥𝑒

∑𝑒∈𝐸

𝑥𝑒=𝑛−1

for all 𝑖 , 𝑗∈𝑉

e

𝑝

ILP for Minimum Routing Cost Tree (Fischetti, L, Serafini)

Given G=(V,E), edge lengths d. Use 0-1 variables for each path

(i)

(ii)

(iii)

min ∑𝑝∈𝑃

𝑑𝑝 𝑥𝑝

=1

∑𝑝∈𝑃 ( 𝑖 , 𝑗 ) :𝑒∈𝑝

𝑥𝑝 ≤𝑥𝑒

∑𝑒∈𝐸

𝑥𝑒=𝑛−1

for all 𝑖 , 𝑗∈𝑉

e

𝑝

Best tree has RC 10% less than RC of best star “ “ “ SP 6% “ “ SP “ “, SP of star is experimentally 10-15% of optimalSP of best tree is 4-10% of optimal