1. Greedy Approximation Algorithms for Covering Problems in Computational Biology
Ion Mandoiu
Computer Science & Engineering Department
University of Connecticut

2. Why Approximation Algorithms?
Most practical optimization problems are NP-hard
Approximation algorithms offer the next best thing to an efficient exact algorithm
Polynomial time
Solutions guaranteed to be “close” to optimum
-approximation algorithm: solution cost within a multiplicative factor of  of optimum cost
Practical relevance: insights needed to establish approximation guarantee often lead to fast, highly effective practical implementations

3. Why Computational Biology?
Exploding multidisciplinary field at the intersection of computer science, biology, discrete mathematics, statistics, optimization, chemistry, physics, …
Source of a fast growing number of combinatorial optimization applications:
TSP and Euler paths in DNA sequencing
Dynamic Programming in sequence alignment
Integer Programming in Haplotype inference …
This talk: two “covering” problems in computational biology (primer set selection and string barcoding)

4. Overview Potential function greedy algorithm
- The set cover problem and the greedy algorithm
- Potential function generalization
Primer Set Selection for Multiplex PCR
The String Barcoding Problem
Conclusions

5. The Set Cover Problem Given: Find: Greedy Algorithm:
Universal set U with n elements
Family of sets (Sx, xX) covering all elements of U
Find:
Minimum size subset X’ of X s.t. (Sx, xX’) covers all elements of U
Greedy Algorithm:
- Start with empty X’, and repeatedly add x such that Sx contains the most uncovered elements until U is covered

6. Approximation Guarantee
Classical result (Johnson’74, Lovasz’75, Chvatal’79): the greedy setcover algorithm has an approximation factor of H(n)=1+1/2+1/3+…+1/n < 1+ln(n)
The approximation factor is tight
Cannot be approximated within a factor of (1-)ln(n) unless NP=DTIME(nloglog(n))

7. General setting “Potential function” (X’)  0 ({}) = max
(X’) = 0 for all feasible solutions
X’’  X’  (X’’)  (X’)
If (X’)>0, then there exists x s.t. (X’+x) < (X’)
X’’  X’  ∆(x,X’)  ∆(x,X’) for every x, where
∆(x,X’) := (X’) - (X’+x)
Problem: find minimum size set X’ with (X’)=0

8. Generic Greedy Algorithm
X’  {}
While (X’) > 0
Find x with maximum ∆(x,X’)
X’  X’ + x
Theorem (Konwar et al.’05) The generic greedy algorithm has an approximation factor of 1+ln ∆max

9. Proof Idea
Let x1, x2,…,xg be the elements selected by greedy, in the order in which they are chosen, and x*1, x*2,…,x*k be the elements of an optimum solution.
Charging scheme: xi charges to x*j a cost of
where ij = ∆(xi,{x1,…, xi-1}{x*1,…,x*j})
Fact 1: Each x*j gets charged a total cost of at most 1+ln ∆max

10. Proof of claim 2
Fact 2: Each xi charges at least 1 unit of cost

11. Overview Potential function greedy algorithm
Primer set selection for multiplex PCR
Motivation and problem formulation
Greedy applied to primer set selection
Experimental results
The String Barcoding Problem
Conclusions

12. DNA Structure Four nucleotide types: A,C,T,G Normally double stranded
A’s paired with T’s
C’s paired with G’s

13. The Polymerase Chain Reaction
Primer 1
Primer 2
Primers
Polymerase
Target Sequence
Repeat cycles

14. Primer Pair Selection Problem
Forward primer
Reverse primer
amplification locus 3' 5'
Given:
Genomic sequence around amplification locus
Primer length k
Amplification upperbound L
Find: Forward and reverse primers of length k that hybridize within a distance of L of each other and optimize amplification efficiency (melting temperature, secondary structure, mis-priming, etc.)

15. Multiplex PCR Multiplex PCR (MP-PCR) Primer set selection
Multiple DNA fragments amplified simultaneously
Each amplified fragment still defined by two primers
A primer may participate in amplification of multiple targets
Primer set selection
Typically done by time-consuming trial and error
An important objective is to minimize number of primers
Reduced assay cost
Higher effective concentration of primers  higher amplification efficiency
Reduced unintended amplification

16. Primer Set Selection Problem
Given:
Genomic sequences around n amplification loci
Primer length k
Amplification upper bound L
Find:
Minimum size set S of primers of length k such that, for each amplification locus, there are two primers in S hybridizing with the forward and reverse genomic sequences within a distance of L of each other

17. Applications Single Nucleotide Polymorphism (SNP) genotyping
Up to thousands of SNPs genotyped simultaneously
Selective PCR amplification required for improved accuracy
Spotted microarray synthesis [Fernandes&Skiena’02]
Primers can be used multiple times
For each target, need a pair of primers amplifying that target and only that target (amplification uniqueness constraint)
Can still reduce #primers from 2n to O(n1/2)

18. Previous Work on Primer Selection
Well-studied problem: [Pearson et al. 96], [Linhart & Shamir’02], [Souvenir et al.’03], etc.
Almost all problem formulations decouple selection of forward and reverse primers
To enforce bound of L on amplification length, select only primers that hybridize within L/2 bases of desired target
In worst case, this method can increase the number of primers by a factor of O(n) compared to the optimum
[Pearson et al. 96] Greedy set cover algorithm gives O(ln n) approximation factor for the “decoupled” formulation

19. Previous Work (contd.)
[Fernandes&Skiena’02] study primer set selection with uniqueness constraints
Minimum Multi-Colored Subgraph Problem:
Vertices correspond to candidate primers
Edge colored by color i between u and v iff corresponding primers hybridize within a distance of L of each other around i-th amplification locus
Goal is to find minimum size set of vertices inducing edges of all colors
Can capture length amplification constraints too

20. Integer Program Formulation
0/1 variable xu for every vertex
0/1 variable ye for every edge e

21. LP-Rounding Algorithm
(1) Solve linear programming relaxation
(2) Select node u with probability xu
(3) Repeat step 2 O(ln(n)) times and return selected nodes
Theorem [Konwar et al.’04]: The LP-rounding algorithm finds a feasible solution at most O(m1/2lnn) times larger than the optimum, where m is the maximum color class size, and n is the number of nodes
For primer selection, m  L2  approximation factor is O(Llnn)
Better approximation?
Unlikely for minimum multi-colored subgraph problem

22. Selection w/o Uniqueness Constraints
Can be seen as a “simultaneous set covering” problem:
- The ground set is partitioned into n disjoint sets Si (one for each target), each with 2L elements
The goal is to select a minimum number of sets (i.e., primers) that cover at least half of the elements in each partition
SNPi L L

23. Greedy Algorithm
Potential function  = minimum number of elements that must be covered = i max{0, L - #uncovered elements in Si}
Initially,  = nL
For feasible solutions,  = 0
∆()  nL (much smaller in practice)
Theorem [Konwar et al.’05]: The number of primers selected by the greedy algorithm is at most 1+ln(nL) larger than the optimum

24. Experimental Setting Datasets extracted from NCBI databases, L=1000
Dell PowerEdge 2.8GHz Xeon
Compared algorithms
G-FIX: greedy primer cover algorithm [Pearson et al.]
MIPS-PT: iterative beam-search heuristic [Souvenir et al.]
Restrict primers to L/2 bases around amplification locus
G-VAR: naïve modification of G-FIX
First selected primer can be up to L bases away
Opposite sequence truncated after selecting first primer
G-POT: potential function driven greedy algorithm

25. Experimental Results, NCBI tests#
Targets k
G-FIX
(Pearson et al.)
G-VAR
(G-FIX with dynamic truncation)
MIPS-PT
(Souvenir et al.)
G-POT
(Potential- function greedy)
#Primers
CPU
sec 20 8 7
0.04
0.08 10 6
0.10 9
0.03 13 15 12 14 18 26
0.11 50
0.13
0.30 21 48
0.32 23
0.22 24
0.36 30
150
0.33 31
0.14 32 41
246 29
0.28
100 17
0.49
0.89
226
0.58 37
0.37
0.72
844
0.75 53
0.59
0.84 75
2601 42
0.61

26. #primers, as percentage of 2n (l=8)

27. #primers, as percentage of 2n (l=10)

28. #primers, as percentage of 2n (l=12)

29. CPU Seconds (l=10) n

30. Overview Potential function greedy algorithm
Primer Set Selection for Multiplex PCR
The String Barcoding Problem
- Problem Formulation
- Integer programming and greedy algorithms
- Experimental results
Conclusions

31. Motivation Rapid pathogen detection
Given
Pathogen with unknown identity
Database of known pathogens
Problem
Identify unknown pathogen quickly
Ideal solution: determine DNA sequence of unknown pathogen

32. Real World Not possible to quickly sequence an unknown pathogen
Only have sequence for pathogens in database
Can quickly test for presence of short substrings in unknown virus (substring tests) using hybridization
String barcoding [Borneman et al.’01, RashGusfield’02]
Use substring tests that uniquely identify each pathogen in the database

33. String Barcoding Problem
Given:
Genomic sequences g1,…, gn
Find:
Minimum number of distinguisher strings t1,…,tk
Such that:
For every gi  gj, there exists a string tl which is substring of gi or gj, but not of both
At least log2n distinguishers needed
Fingerprints  n distinguishers

34. Example Given sequences: Feasible set of distinguishers: {tg, atgga}
1. cagtgc
2. cagttc
3. catgga
Feasible set of distinguishers: {tg, atgga} tg
atgga
cagtgc 1
cagttc
catgga
Row vectors: unique barcodes for each pathogen

35. Computational Complexity
[Berman et al.’04] Cannot be approximated within a factor of (1-)ln(n) unless NP=DTIME(nloglog(n))

36. Setcover Greedy Algorithm
Distinguisher selection as setcover problem
Elements to be covered are the pairs of sequences
Each candidate distinguisher defines a set of pairs that it separates
Another view: covering all edges of a complete graph with n vertices by the minimum number of given cuts
For n sequences, largest set can have O(n2) elements
 The setcover greedy guarantees ln(n2) = 2 ln n approximation

37. Integer Program Formulation
0/1 variable for each candidate distinguisher
1  candidate is selected
0  candidate is not selected
For each pair of sequences, at least one candidate separating them is selected
Objective Function
Minimize #selected candidates

38. Practical Issues Quadratic # of constraints, huge # of variables
Genome sizes range from thousands of bases for phage and viruses to millions for bacteria to billions for higher organisms
Many variables can be removed:
Candidates that appear in all sequences
Sufficient to keep a single candidate among those that appear in the same set of sequences
How to efficiently remove useless variables?
Rash&Gusfield use suffix trees

39. Suffix Tree Example Strings: 1. cagtgc 2. cagttc 3. catgga
v1 - {1,2,3}
v2 - {1,2,3}
v3 - {3}
v4 - {1}
v5 - {3}
v6 - {1,2}
v7 - {2}
v8 - {1}
v9 - {1,2,3}
v10 - {1,2,3}
v11 - {1,2}
v12 - {1}
v13 - {2}
v14 - {3}
v15 - {1,2,3}
v16 - {2}
v17 - {2}
v18 - {1,3}
v19 - {1}
v20 - {3}
v21 - {1,2,3}
v22 - {3}
v23 - {2}
v24 - {1,2}
v25 - {1}

40. Integer Program Minimize
V18 + V22 + V11 + V17 + V8 #objective function
Such that
V18 + V17 + V8 >= 1 #constraint to cover pair 1,2
V22 + V11 + V8 >= 1 #constraint to cover pair 1,3
V18 + V22 + V11 + V17 >= 1 #constraint to cover pair 2,3
Binaries #all variables are 0/1
V18 V22 V11 V17 V8
End
tg (V18)
atgga (V22)
cagtgc 1
cagttc
catgga

41. Limitations of Integer Program Method
Works only for small instances
sequences
Average length ~1000 characters
Over 4 hours needed to come within 20% of optimum!
Scalable Heuristics?

42. Distinguisher Induced Partition
Key idea [Berman et al. 04]: Keep track of the partition defined by distinguishers selected so far
Distinguisher 1 1 2 3
n-1 n
Distinguisher 2

43. Information Content Heuristic
 = partition entropy = log2(#permutations compatible with current partition)
Initial partition entropy = log2(n!)  n log2n
For feasible distinguisher sets, partition entropy = 0
∆()  n :
log2(n!) - log2(k!(n-k)!) < log2(2n) = n
Information content heuristic (ICH) = greedy driven by partition entropy
Theorem [Berman et al.’04] ICH has an approximation factor of 1+ln(n)

44. ICH Limitations Real genomic data has degenerate nucleotides
Ambiguous sequencing
Single nucleotide polymorphisms
For sequences with degenerate nucleotides there are three possibilities for distinguisher hybridization
Sure hybridization
Sure mismatch
Uncertain hybridization
 No partition to work with!

45. Practical Implementation
ICH and setcover greedy give nearly identical results on data w/o non-degenerate bases
Setcover greedy can also be extended to handle
degenerate bases in the sequences
redundancy requirements (each pair of sequences must be separated r times)
Two main steps for both algorithms:
Candidate generation
Greedy selection

46. Candidate Generation Can be done using suffix trees
We use a simpler yet efficient incremental approach
Candidates that match all or only one sequence are removed from consideration
Solution quality is similar even when candidates are generated from a single sequence
Equivalent to considering only distinguisher sets that assign a barcode of (1,1,…,1) to the source sequence

47. Candidate Selection Evaluate ∆() for all candidates and choose best
Speed-up techniques
Efficient gain computation using partition data-structure
Lazy gain update: if old ∆() is lower than best so far, do not recompute

48. Experimental Results
mat mat part part #
n lazy lazy dist
Averages over 10 testcases, sequence length = 10,000
Barcodes for 100 sequences of length 1,000,000 computed in less than 10 minutes

49. Overview Potential function greedy algorithm
Primer Set Selection for Multiplex PCR
The String Barcoding Problem
Conclusions

50. Conclusions
General potential function framework for designing and analyzing greedy covering algorithms
Improved approximation guarantees and practical performance for two important optimization problems in computational biology: primer set selection for multiplex PCR, and distinguisher selection for string barcoding

51. Ongoing Work Primer Set Selection String Barcoding
Improved hybridization models
Degenerate primers
Partitioning into multiple multiplexed PCR reactions
Close approximation gap for minimum multicolored sub-graph
String Barcoding
Probe mixtures as distinguishers
Beyond redundancy: error correcting
Simultaneous detection of multiple pathogens

52. Acknowledgments B. DasGupta, K. Konwar, A. Russell, A. Shvartsman
UCONN Research Foundation