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

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

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

DNA Physical Mapping[9] m2 m1 m5 m4 m3

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

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)

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

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?

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?

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

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

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 ;

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

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

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

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

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]

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.

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

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

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.

References 1. N.S. Narayanaswamy , R. Subashini , “A new characterization of matirces with Consecutive Ones Propery”, Discrete Applied Mathematics, August 2009. 2. 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,1976. 3. M.C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs”, Academic Press, 1980. 4. R.Wang, F.C.M Lau, Y.C. Zhao, “Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices”, Discrete Applied Mathematics, 2007. 5. S. Ghosh, “File organization: The consecutive retrieval property”, Communications of the ACM,1979. 6. D. Fulkerson, O.A Gross, “Incidence matrices and interval graphs”, Pacific Journal of Mathematics,1965.

7. A. Tucker, “A structure theorem for the consecutive ones property”, Journal of Combinatorial Theory, 1972. 8. J. Meidanis, E. Munuera, “A theory for the consecutive ones property”, Proceedings of the III South American Workshop on String Processing, 1996. 9. J. Meidanis, Oscar Porto, Guilherme P. Telles, “On the Consecutive Ones Property”, Discrete Applied Mathematics, 1998.