1. Characterizing Matrices with Consecutive Ones Property
N.S. Narayanaswamy, IIT Madras
(Joint work with R. Subashini, NITC)

2. The Problem
Does a given 0-1 Matrix have the Consecutive Ones Property (COP)?

3. Consecutive Ones Property
Permute the rows such that the ones in each column occur consecutively

4. DNA Physical Mapping[9] m2 m1 m5 m4 m3

5. Applications Maximal Clique-Vertex incidence matrix
Interval graph characterization
Characterizing cubic Hamiltonian graphs
Databases
Computational Biology

6. Previous Work Poly time solvable
Fulkerson and Gross
Forbidden matrix configurations – Tucker
View the matrix as a maximal clique-vertex incidence matrix
Asteroidal triple
Induced cycles larger than K3
Linear time algorithm
Booth and Lueker
Running time of O(m+n+#non-zero entries)

7. Consecutive Ones Testing (COT) Trees
PQ-trees
L-R order yields one permutation
Leaves are the rows
Internal nodes are P and Q nodes
P node - all permutations of its children yields a valid permutation
Q node - exactly two permutations are permitted
Algorithm outputs a PQ-tree only if the matrix has the COP
Addressed by PQR trees

8. Permutations and Intervals
A feasible permutation of rows yields an interval assignment to the columns
Length of the interval is the number of ones in the column
Intersection cardinality of a pair of intervals is the number of rows in which a 1 occurs in both the corresponding columns.
Does such an assignment imply a feasible permutation?

9. Feasible Interval Assignments
Each Permutation gives a interval assignment
Is it sufficient to find an interval assignment to the sets to preserve intersection cardinalities
If yes, can we get a permutation from an interval assignment?

10. Preserving intersection cardinalities - Sufficiency
Sort the intervals in increasing order of left end point and break ties using the right end points
Discard identical columns
Consider (P1,Q1)
Pi row indices in i-th column
Qi is the interval assigned to the i-th column
Encodes all permutations in which Pi is mapped to Qi

11. Refining the set of permutations
Iteratively filter the current set of permutations
Using strictly intersecting pairs
Pair of intersecting intervals, neither contained in the other

12. Algorithm 1 - Permutations from an ICPIA
Let 0 = {(Ai, Bi) | 1 ≤ i ≤ m} j = 1; while there is (P1,Q1), (P2,Q2) Є  j-1 with Q1 and Q2 strictly intersecting do  j =  j-1\{(P1,Q1), (P2, Q2)}  j =  j  {(P1  P2 ,Q1  Q2),(P1\ P2, Q1\ Q2),(P2\ P1 ,Q2\ Q1)} j= j+1; end while = j Return ;

13. Proof Helly property for intervals Intersection cardinality preserved
For any 3 mutually intersecting intervals one is contained in the union of the other two.
Intersection cardinality preserved

14. Invariants Q is an interval for each (P,Q). |P|=|Q| for each (P,Q)
For any two (P',Q'), (P'',Q''),
|P'P''|=|Q'  Q''|.
At the end no interval is strictly intersecting with another interval
Either disjoint or contained.

15. Completing the refinement
The set of (P,Q) yields a natural containment tree

16. Algorithm 2 – Permutations from Algorithm 1
function Post-order-traversal(T, root-node, ) if (root-node is a leaf) then return endif while (root-node has unexplored children) do Next-root-node = an unexplored child of root node Post-order-traversal(T, next-root-node, П) end while if (root-node has no unexplored children) then let (P,Q) denote the element of П associated with the root-node let (P1,Q1)…(Pk,Qk) be the pairs associated with the children of root-node П = П\{(P,Q)} П = П U {(P\(P1U …Pk), Q\(Q1U…. Qk))} Return

17. Consequence Given an interval assignment
We have a data structure that encodes all permutations which yield this interval assignment

18. Finding good interval assignments
For a set of proper intervals and its flipping the intersection graph are isomorphic- [1,8],[5,10],[2,7] is isomorphic to [1,6],[3,10],[4,9]

19. Feasible interval assignments
Intuition
To assign intervals to a set system, there are only two choices and these will be decided at the first step.

20. An ordering of the sets First set Next Set (iteratively)
A set such that all those sets which intersect it have a pair-wise non-empty intersection - candidate for the leftmost interval
Next Set (iteratively)
One that has a strict intersection with one of the chosen sets.

21. Assigning the Intervals
First Set-Left most interval
Second set - has strict intersection with first set. So two interval choices
Next set (iteratively)-has strict intersection with some interval
Exactly one choice of interval, given intersection cardinality constraints
Failure implies no feasible interval assignment
Linear time in the number of sets, but computing intersection is costly

22. Sets left out Do not have a strict overlap with the sets considered
Disjoint
Contained
Two distinct sets are related if they have a strict overlap
Consider connected components in this undirected graph

23. On the components Each component is a sub-matrix formed by the columns
Two components are either
Disjoint
Or all the sets in one are contained in a single set of the other.
An interval assignment to each component implies an interval assignment to the whole set system

24. Putting the interval assignments together
Given that an interval assignment to each of the components is feasible.
Containment tree/forest on the components
An arc between vertices corresponding to two components if the sets of one are all contained in one set of the other
Construct the interval assignment in a BFS fashion starting from the root of each tree

25. Applications Can test if rows can be permuted
so that columns are sorted
1s occur in a circular fashion

26. Further Research
Solves an isomorphism problem to a target class of matrices in which 1s in each column are consecutive
NP-hard when 1s are in at most 3 consecutive regions.

27. References
1. N.S. Narayanaswamy , R. Subashini , “A new characterization of matirces with Consecutive Ones Propery”, Discrete Applied Mathematics, August K.S. Booth, G.S. Lueker, “Tesing for the consecutive ones property, interval graph and graph planarity using PQ-tree algorithms”, Journal of computer System Science, M.C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs”, Academic Press, R.Wang, F.C.M Lau, Y.C. Zhao, “Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices”, Discrete Applied Mathematics, S. Ghosh, “File organization: The consecutive retrieval property”, Communications of the ACM, D. Fulkerson, O.A Gross, “Incidence matrices and interval graphs”, Pacific Journal of Mathematics,1965.

28. 7. A. Tucker, “A structure theorem for the consecutive ones property”, Journal of Combinatorial Theory, J. Meidanis, E. Munuera, “A theory for the consecutive ones property”, Proceedings of the III South American Workshop on String Processing, J. Meidanis, Oscar Porto, Guilherme P. Telles, “On the Consecutive Ones Property”, Discrete Applied Mathematics, 1998.