On the Gapped Consecutive-Ones Property Cedric Chauve, J an Manuch - - PowerPoint PPT Presentation

Introduction and results statement Technical details Conclusion

On the Gapped Consecutive-Ones Property

Cedric Chauve, J´ an Manuch and Murray Patterson

• Dept. Mathematics and School of Computing Science, Simon Fraser University,

Canada

EuroComb 2009, September 7th, 2009

Work funded by NSERC and the France-Canada Scientific Cooperation.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped Consecutive-Ones Property

Definitions.

• 1. Let M be a binary matrix. A gap of length δ in a row of M is

a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped Consecutive-Ones Property

Definitions.

• 1. Let M be a binary matrix. A gap of length δ in a row of M is

a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row.

• 2. M is k-C1P if there exists a total order of its columns such

that each row contains at most k − 1 gaps.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped Consecutive-Ones Property

Definitions.

• 1. Let M be a binary matrix. A gap of length δ in a row of M is

a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row.

• 2. M is k-C1P if there exists a total order of its columns such

that each row contains at most k − 1 gaps.

• 3. M is (k, δ)-C1P if it is k-C1P and each gap is of length at

most δ.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped Consecutive-Ones Property

Definitions.

• 1. Let M be a binary matrix. A gap of length δ in a row of M is

a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row.

• 2. M is k-C1P if there exists a total order of its columns such

that each row contains at most k − 1 gaps.

• 3. M is (k, δ)-C1P if it is k-C1P and each gap is of length at

most δ.

Example: a (2, 1)-C1P matrix of maximum degree 3.

1 1 1 1 1 1 1 1

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped C1P: Problem and Our Results Summary

Problem.

What is the complexity of deciding if, given a binary matrix M, k ≥ 2 and δ ≥ 1, M is a (k, δ)-C1P matrix ?

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped C1P: Problem and Our Results Summary

Problem.

What is the complexity of deciding if, given a binary matrix M, k ≥ 2 and δ ≥ 1, M is a (k, δ)-C1P matrix ?

Theorem 1.

Deciding if a binary matrix M is (k, δ)-C1P is NP-complete for k, δ ≥ 2.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped C1P: Problem and Our Results Summary

Problem.

What is the complexity of deciding if, given a binary matrix M, k ≥ 2 and δ ≥ 1, M is a (k, δ)-C1P matrix ?

Theorem 1.

Deciding if a binary matrix M is (k, δ)-C1P is NP-complete for k, δ ≥ 2.

Theorem 2.

Deciding if a binary matrix M is (k, 1)-C1P is NP-complete for k ≥ 3.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

The Gapped C1P: Problem and Our Results Summary

Problem.

What is the complexity of deciding if, given a binary matrix M, k ≥ 2 and δ ≥ 1, M is a (k, δ)-C1P matrix ?

Theorem 1.

Deciding if a binary matrix M is (k, δ)-C1P is NP-complete for k, δ ≥ 2.

Theorem 2.

Deciding if a binary matrix M is (k, 1)-C1P is NP-complete for k ≥ 3.

Theorem 3.

Deciding if an m × n binary matrix M of maximum degree s is (k, δ)-C1P can be done in O(nms+(k−1)δ) worst-case time and space.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Motivation: paleogenomics

Reconstructing ancestral genomes (that can not be sequenced due to DNA decay) from common characters of current genomes (Chauve and Tannier 2008).

chicken human mouse dog

...

2

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Motivation: paleogenomics

• 1. Binary matrix can be used to encode possible genome

segments of an unknown ancestral genome:

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Motivation: paleogenomics

• 1. Binary matrix can be used to encode possible genome

segments of an unknown ancestral genome:

◮ Columns represent “genes” that were present in the ancestral

genome.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Motivation: paleogenomics

• 1. Binary matrix can be used to encode possible genome

segments of an unknown ancestral genome:

◮ Columns represent “genes” that were present in the ancestral

genome.

◮ Rows represent groups of genes that should be co-localized.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Motivation: paleogenomics

• 1. Binary matrix can be used to encode possible genome

segments of an unknown ancestral genome:

◮ Columns represent “genes” that were present in the ancestral

genome.

◮ Rows represent groups of genes that should be co-localized.

• 2. Ordering columns correspond to ordering genes along

chromosomes of the unknown ancestral genome.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Motivation: paleogenomics

• 1. Binary matrix can be used to encode possible genome

segments of an unknown ancestral genome:

◮ Columns represent “genes” that were present in the ancestral

genome.

◮ Rows represent groups of genes that should be co-localized.

• 2. Ordering columns correspond to ordering genes along

chromosomes of the unknown ancestral genome.

• 3. Strict combinatorial framework: C1P=(1, 0)-C1P (Chauve

and Tannier 2008).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Motivation: paleogenomics

• 1. Binary matrix can be used to encode possible genome

segments of an unknown ancestral genome:

◮ Columns represent “genes” that were present in the ancestral

genome.

◮ Rows represent groups of genes that should be co-localized.

• 2. Ordering columns correspond to ordering genes along

chromosomes of the unknown ancestral genome.

• 3. Strict combinatorial framework: C1P=(1, 0)-C1P (Chauve

and Tannier 2008).

• 4. Relaxed combinatorial framework to account for evolutionary

noise (missing genes, lineage-specific rearrangements, . . . ): (k, δ)-C1P with small values for k and δ.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Link with the Graph bandwidth

Definition.

A graph G = (V , E) has bandwidth b if its vertices can be

• rdered, say v1 . . . vn, such that for every edge (vi, vj), |j − i| ≤ b.
• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Link with the Graph bandwidth

Definition.

A graph G = (V , E) has bandwidth b if its vertices can be

• rdered, say v1 . . . vn, such that for every edge (vi, vj), |j − i| ≤ b.

Matrices and graphs.

A graph G has bandwidth b if and only if its adjacency matrix MG (of maximum degree 2) is (2, b − 1)-C1P.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Link with the Graph bandwidth

Definition.

A graph G = (V , E) has bandwidth b if its vertices can be

• rdered, say v1 . . . vn, such that for every edge (vi, vj), |j − i| ≤ b.

Matrices and graphs.

A graph G has bandwidth b if and only if its adjacency matrix MG (of maximum degree 2) is (2, b − 1)-C1P.

Known results on the Graph Bandwidth.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Link with the Graph bandwidth

Definition.

A graph G = (V , E) has bandwidth b if its vertices can be

• rdered, say v1 . . . vn, such that for every edge (vi, vj), |j − i| ≤ b.

Matrices and graphs.

A graph G has bandwidth b if and only if its adjacency matrix MG (of maximum degree 2) is (2, b − 1)-C1P.

Known results on the Graph Bandwidth.

• 1. Bandwidth Minimization is NP-complete (Garey et al. 1978).
• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Link with the Graph bandwidth

Definition.

A graph G = (V , E) has bandwidth b if its vertices can be

• rdered, say v1 . . . vn, such that for every edge (vi, vj), |j − i| ≤ b.

Matrices and graphs.

A graph G has bandwidth b if and only if its adjacency matrix MG (of maximum degree 2) is (2, b − 1)-C1P.

Known results on the Graph Bandwidth.

• 1. Bandwidth Minimization is NP-complete (Garey et al. 1978).
• 2. Deciding if a graph has bandwidth b can be done in O(nb+1)

time and space (Saxe 1980).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Link with the Graph bandwidth

Definition.

A graph G = (V , E) has bandwidth b if its vertices can be

• rdered, say v1 . . . vn, such that for every edge (vi, vj), |j − i| ≤ b.

Matrices and graphs.

A graph G has bandwidth b if and only if its adjacency matrix MG (of maximum degree 2) is (2, b − 1)-C1P.

Known results on the Graph Bandwidth.

• 1. Bandwidth Minimization is NP-complete (Garey et al. 1978).
• 2. Deciding if a graph has bandwidth b can be done in O(nb+1)

time and space (Saxe 1980).

• 3. Deciding if a graph has bandwidth 2 can be done in linear

time (Caprara et al. 2002).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Background on the (gapped) C1P

Deciding C1P.

Deciding if a matrix is C1P can be done in linear time (Booth and Lueker 1976, . . . , McConnell 2004).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Background on the (gapped) C1P

Deciding C1P.

Deciding if a matrix is C1P can be done in linear time (Booth and Lueker 1976, . . . , McConnell 2004).

Deciding k-C1P.

Deciding if a matrix is k-C1P is NP-complete for k ≥ 2 (Goldberg et al. 1995).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Background on the (gapped) C1P

Deciding C1P.

Deciding if a matrix is C1P can be done in linear time (Booth and Lueker 1976, . . . , McConnell 2004).

Deciding k-C1P.

Deciding if a matrix is k-C1P is NP-complete for k ≥ 2 (Goldberg et al. 1995).

Matrices of maximum degree 2 (graphs).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Background on the (gapped) C1P

Deciding C1P.

Deciding if a matrix is C1P can be done in linear time (Booth and Lueker 1976, . . . , McConnell 2004).

Deciding k-C1P.

Deciding if a matrix is k-C1P is NP-complete for k ≥ 2 (Goldberg et al. 1995).

Matrices of maximum degree 2 (graphs).

• 1. Minimizing the number of gaps is NP-complete (Haddadi

2002).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Background on the (gapped) C1P

Deciding C1P.

Deciding if a matrix is C1P can be done in linear time (Booth and Lueker 1976, . . . , McConnell 2004).

Deciding k-C1P.

Deciding if a matrix is k-C1P is NP-complete for k ≥ 2 (Goldberg et al. 1995).

Matrices of maximum degree 2 (graphs).

• 1. Minimizing the number of gaps is NP-complete (Haddadi

2002).

• 2. Minimizing the length of gaps is NP-complete (Graph

Bandwidth Minimization).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proofs of NP-completeness: principle

• 1. We prove NP-completeness of (k, δ)-C1P by reduction from

3SAT.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proofs of NP-completeness: principle

• 1. We prove NP-completeness of (k, δ)-C1P by reduction from

3SAT.

• 2. From a 3CNF φ with n variables and m clauses, we define a

matrix Mφ with O(n + m + k) rows and columns that is (k, δ)-C1P if and only if φ is statisfiable.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proofs of NP-completeness: principle

• 1. We prove NP-completeness of (k, δ)-C1P by reduction from

3SAT.

• 2. From a 3CNF φ with n variables and m clauses, we define a

matrix Mφ with O(n + m + k) rows and columns that is (k, δ)-C1P if and only if φ is statisfiable.

• 3. Main idea (from Goldberg et al. 1995).
• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proofs of NP-completeness: principle

• 1. We prove NP-completeness of (k, δ)-C1P by reduction from

3SAT.

• 2. From a 3CNF φ with n variables and m clauses, we define a

matrix Mφ with O(n + m + k) rows and columns that is (k, δ)-C1P if and only if φ is statisfiable.

• 3. Main idea (from Goldberg et al. 1995).

3.1 Associate each variable (resp. clause Cj) vi of φ to a specific set of columns bi (resp. Bj) and an order inside this block linked to the value assigned to vi.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proofs of NP-completeness: principle

• 1. We prove NP-completeness of (k, δ)-C1P by reduction from

3SAT.

• 2. From a 3CNF φ with n variables and m clauses, we define a

matrix Mφ with O(n + m + k) rows and columns that is (k, δ)-C1P if and only if φ is statisfiable.

• 3. Main idea (from Goldberg et al. 1995).

3.1 Associate each variable (resp. clause Cj) vi of φ to a specific set of columns bi (resp. Bj) and an order inside this block linked to the value assigned to vi. 3.2 Create a block D of columns that will have its order fixed to have almost k gaps in some row.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proofs of NP-completeness: principle

• 1. We prove NP-completeness of (k, δ)-C1P by reduction from

3SAT.

• 2. From a 3CNF φ with n variables and m clauses, we define a

matrix Mφ with O(n + m + k) rows and columns that is (k, δ)-C1P if and only if φ is statisfiable.

• 3. Main idea (from Goldberg et al. 1995).

3.1 Associate each variable (resp. clause Cj) vi of φ to a specific set of columns bi (resp. Bj) and an order inside this block linked to the value assigned to vi. 3.2 Create a block D of columns that will have its order fixed to have almost k gaps in some row. 3.3 Associate clause Cj of φ involving variables vp, vq, vr to rows in such a way that, one of these row will not be (k, δ)-C1P due to the gaps in D and the gaps in bp, bq, br and Bj if and only if vp, vq and vr are false.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

An important lemma

Lemma (C., Manuch and Patterson 2009).

For k ≥ 2, δ ≥ 1 and n ≥ 2δ + 3, given a matrix M on N ≥ n columns, by adding n(δ + 1) − δ(δ+3)

2

− 1 rows to M, it is possible to force n selected columns to appear contiguous in a fixed order (up to a reversal) in any (k, δ)-C1P ordering of the columns of M.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

An important lemma

Lemma (C., Manuch and Patterson 2009).

For k ≥ 2, δ ≥ 1 and n ≥ 2δ + 3, given a matrix M on N ≥ n columns, by adding n(δ + 1) − δ(δ+3)

2

− 1 rows to M, it is possible to force n selected columns to appear contiguous in a fixed order (up to a reversal) in any (k, δ)-C1P ordering of the columns of M.

Use of this result.

Given a matrix M add rows that fix the order of a set of columns that contain a given number (close to k) of gaps of length at most δ.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

An important lemma: sketch of proof

The augmented matrix.

Let M be the original matrix and C = {i + 1, . . . , i + n} be the columns we want to fix in the order i + 1 . . . i + n (i ≤ N − n). We add rows {i, j} to M for any 1 ≤ i < j ≤ n such that |i − j| ≤ δ + 1.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

An important lemma: sketch of proof

The augmented matrix.

Let M be the original matrix and C = {i + 1, . . . , i + n} be the columns we want to fix in the order i + 1 . . . i + n (i ≤ N − n). We add rows {i, j} to M for any 1 ≤ i < j ≤ n such that |i − j| ≤ δ + 1.

Three important steps.

Let π be an ordering of the columns of the augmented matrix.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

An important lemma: sketch of proof

The augmented matrix.

Let M be the original matrix and C = {i + 1, . . . , i + n} be the columns we want to fix in the order i + 1 . . . i + n (i ≤ N − n). We add rows {i, j} to M for any 1 ≤ i < j ≤ n such that |i − j| ≤ δ + 1.

Three important steps.

Let π be an ordering of the columns of the augmented matrix.

• 1. If columns π(i) and π(j) of C are close in M but not in π,

then π is not a (k, δ)-C1P ordering.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

An important lemma: sketch of proof

The augmented matrix.

Let M be the original matrix and C = {i + 1, . . . , i + n} be the columns we want to fix in the order i + 1 . . . i + n (i ≤ N − n). We add rows {i, j} to M for any 1 ≤ i < j ≤ n such that |i − j| ≤ δ + 1.

Three important steps.

Let π be an ordering of the columns of the augmented matrix.

• 1. If columns π(i) and π(j) of C are close in M but not in π,

then π is not a (k, δ)-C1P ordering.

• 2. If columns of C are not contiguous in π then π is not a

(k, δ)-C1P ordering.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

An important lemma: sketch of proof

The augmented matrix.

Let M be the original matrix and C = {i + 1, . . . , i + n} be the columns we want to fix in the order i + 1 . . . i + n (i ≤ N − n). We add rows {i, j} to M for any 1 ≤ i < j ≤ n such that |i − j| ≤ δ + 1.

Three important steps.

Let π be an ordering of the columns of the augmented matrix.

• 1. If columns π(i) and π(j) of C are close in M but not in π,

then π is not a (k, δ)-C1P ordering.

• 2. If columns of C are not contiguous in π then π is not a

(k, δ)-C1P ordering.

• 3. If M = C, then for all i ∈ {1, . . . , δ + 1, n − δ, . . . , n} either

π(i) = i or π(i) = n − i + 1.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

◮ Columns 2n + 1 . . . , 2n + 2k fixed to create k − 2 gaps (gaps

in columns D enforced by previous lemma).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

◮ Columns 2n + 1 . . . , 2n + 2k fixed to create k − 2 gaps (gaps

in columns D enforced by previous lemma).

◮ Clause Cj associated to block of 5 columns

Bj = {B1

j = 2n + 2k + 5j − 4, . . . , B5 j = 2n + 2k + 5j}.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

◮ Columns 2n + 1 . . . , 2n + 2k fixed to create k − 2 gaps (gaps

in columns D enforced by previous lemma).

◮ Clause Cj associated to block of 5 columns

Bj = {B1

j = 2n + 2k + 5j − 4, . . . , B5 j = 2n + 2k + 5j}. ◮ Assume Cj involves variables vp, vq and vr and add rows

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

◮ Columns 2n + 1 . . . , 2n + 2k fixed to create k − 2 gaps (gaps

in columns D enforced by previous lemma).

◮ Clause Cj associated to block of 5 columns

Bj = {B1

j = 2n + 2k + 5j − 4, . . . , B5 j = 2n + 2k + 5j}. ◮ Assume Cj involves variables vp, vq and vr and add rows

• 1. that force a gap in bp and B1

j to be first in Bj if vp is false,

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

◮ Columns 2n + 1 . . . , 2n + 2k fixed to create k − 2 gaps (gaps

in columns D enforced by previous lemma).

◮ Clause Cj associated to block of 5 columns

Bj = {B1

j = 2n + 2k + 5j − 4, . . . , B5 j = 2n + 2k + 5j}. ◮ Assume Cj involves variables vp, vq and vr and add rows

• 1. that force a gap in bp and B1

j to be first in Bj if vp is false,

• 2. that force B5

j to be last in Bj if vr is false,

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

◮ Columns 2n + 1 . . . , 2n + 2k fixed to create k − 2 gaps (gaps

in columns D enforced by previous lemma).

◮ Clause Cj associated to block of 5 columns

Bj = {B1

j = 2n + 2k + 5j − 4, . . . , B5 j = 2n + 2k + 5j}. ◮ Assume Cj involves variables vp, vq and vr and add rows

• 1. that force a gap in bp and B1

j to be first in Bj if vp is false,

• 2. that force B5

j to be last in Bj if vr is false,

• 3. that force B3

j to be third in Bj if vq is false,

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 1 (k, δ ≥ 2)

◮ Variable vi associated to block of columns bi = {2i − 1, 2i}

(ordered 2i − 1, 2i if vi is true).

◮ Columns 2n + 1 . . . , 2n + 2k fixed to create k − 2 gaps (gaps

in columns D enforced by previous lemma).

◮ Clause Cj associated to block of 5 columns

Bj = {B1

j = 2n + 2k + 5j − 4, . . . , B5 j = 2n + 2k + 5j}. ◮ Assume Cj involves variables vp, vq and vr and add rows

• 1. that force a gap in bp and B1

j to be first in Bj if vp is false,

• 2. that force B5

j to be last in Bj if vr is false,

• 3. that force B3

j to be third in Bj if vq is false,

• 4. {D, B1, . . . , Bj−1, B1

j , B3 j , B5 j }.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Proof of Theorem 2 (k ≥ 3, δ = 1)

Same principle.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Algorithmic result

Theorem 3.

Deciding if an m × n binary matrix M of maximum degree s is (k, δ)-C1P can be done in O(nms+(k−1)δ) worst-case time and space.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Algorithmic result

Theorem 3.

Deciding if an m × n binary matrix M of maximum degree s is (k, δ)-C1P can be done in O(nms+(k−1)δ) worst-case time and space.

Outline.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Algorithmic result

Theorem 3.

Deciding if an m × n binary matrix M of maximum degree s is (k, δ)-C1P can be done in O(nms+(k−1)δ) worst-case time and space.

Outline.

• 1. Define a graph GM such that, if M is (k, δ)-C1P, then GM has

bandwidth at most s + (k − 1)δ − 1.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Algorithmic result

Theorem 3.

Deciding if an m × n binary matrix M of maximum degree s is (k, δ)-C1P can be done in O(nms+(k−1)δ) worst-case time and space.

Outline.

• 1. Define a graph GM such that, if M is (k, δ)-C1P, then GM has

bandwidth at most s + (k − 1)δ − 1.

• 2. Use a modification of a relatively brute-force algorithm by

(Saxe 1980) to test graph bandwidth.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Open problems

• 1. Complexity of deciding the (2, 1)-C1P.
• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Open problems

• 1. Complexity of deciding the (2, 1)-C1P.
• 2. Encoding the set of all (k, δ)-C1P orderings (extending

PQ-trees and PQR-trees to the Gapped C1P Problem).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Open problems

• 1. Complexity of deciding the (2, 1)-C1P.
• 2. Encoding the set of all (k, δ)-C1P orderings (extending

PQ-trees and PQR-trees to the Gapped C1P Problem).

• 3. A forbidden minor characterization of non (k, δ)-C1P
• matrices. Exists for the C1P (Tucker 1972).
• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Open problems

• 1. Complexity of deciding the (2, 1)-C1P.
• 2. Encoding the set of all (k, δ)-C1P orderings (extending

PQ-trees and PQR-trees to the Gapped C1P Problem).

• 3. A forbidden minor characterization of non (k, δ)-C1P
• matrices. Exists for the C1P (Tucker 1972).
• 4. Considering matrices of bounded degree (pertinent from a

paleogenomics point of view):

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Open problems

• 1. Complexity of deciding the (2, 1)-C1P.
• 2. Encoding the set of all (k, δ)-C1P orderings (extending

PQ-trees and PQR-trees to the Gapped C1P Problem).

• 3. A forbidden minor characterization of non (k, δ)-C1P
• matrices. Exists for the C1P (Tucker 1972).
• 4. Considering matrices of bounded degree (pertinent from a

paleogenomics point of view):

◮ Complexity of deciding the (k, δ)-C1P.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

Open problems

• 1. Complexity of deciding the (2, 1)-C1P.
• 2. Encoding the set of all (k, δ)-C1P orderings (extending

PQ-trees and PQR-trees to the Gapped C1P Problem).

• 3. A forbidden minor characterization of non (k, δ)-C1P
• matrices. Exists for the C1P (Tucker 1972).
• 4. Considering matrices of bounded degree (pertinent from a

paleogenomics point of view):

◮ Complexity of deciding the (k, δ)-C1P. ◮ Efficient alternatives to the brute-force algorithm. Exists for

maximum degree 2 and δ = 1 (Caprara et al. 2002).

• C. Chauve et al.

On the Gapped Consecutive-Ones Property

Introduction and results statement Technical details Conclusion

References

◮ Tucker. 1972. J. Combinat. Theory 12:153-162. ◮ Booth and Lueker. 1976. J. Comput. Syst. Sci. 13:335-379. ◮ Garey, Graham, Johnson and Knuth. 1978. SIAM J. Appl. Math.

34:477-495.

◮ Saxe. 1980. SIAM J. Algebraic Discrete Methods 1:363-369. ◮ Goldberg, Golumbic, Kaplan and Shamir. 1995. J. Comput. Biol.

2:139-152.

◮ Caprara, Malucelli and Petrolani. 2002. Discrete Appl. Math.

117:1-13.

◮ Haddadi. 2002. Int. Trans. Oper. Res. 9:775-777. ◮ McConnell. 2004. ACM-SIAM SODA 15:761-770. ◮ Chauve and Tannier. 2008. PLoS Comput. Biol. 4:e1000234.

• C. Chauve et al.

On the Gapped Consecutive-Ones Property