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

2. The Problem Does a given 0-1 Matrix have the Consecutive Ones Property
Permute the rows such that the ones in each column occur consecutively
Application
Maximal Clique-Vertex incidence matrix
Interval graph characterization
Characterizing cubic Hamiltonian graphs
Databases and Computational Biology

3. Status Poly time solvable - forbidden matrix configurations - Tucker
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 Leuker
Running time of O(m+n+#non-zero entries)

4. 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 permitted
Algorithm outputs a PQ-tree only if the matrix has the COP
Addressed by PQR trees

5. 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?

6. Preserving intersection cardinalities is sufficient
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

7. 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

8. 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.

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

10. Consequence Given an interval assignment Proof uses
We have a data structure that encodes all permutations which yield this interval assignment
Proof uses
Helly property for intervals
For any 3 mutually intersecting intervals one is contained in the union of the other two.
Intersection cardinality preserved

11. 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]
Intuition
To assign intervals to a set system, there are only two choices and these will be decided at the first step.

12. 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.

13. 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

14. 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

15. 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

16. 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

17. Comments Can test if rows can be permuted Recent thoughts
so that columns are sorted
1s occur in a circular fashion
Recent thoughts
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.