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

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

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)

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

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?

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

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

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.

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

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

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.

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.

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

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

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

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

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.