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

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 1function 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

endif

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

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.

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