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Jump Number of Two-Directional Orthogonal RayGraphs

Jose A. Soto1 Claudio Telha2

1Department of Mathematics, MIT

2Operations Research Center, MIT

IPCO 2011

Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 1

Outline

1 Jump Number Problem and 2DORGs

2 Cross-free Matchings and Biclique Covers

3 Main results

Jump Number Problem

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P

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4 Jumps

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P

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4 Jumps 3 Jumps

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P

Jump number of a poset PFind a linear extension (total order)with minimum number of jumps j(P).


(cont.) Jump number

Properties of j(P)

[Habib 84] Comparability invariant: j(G).

[Muller 90] NP-hard for chordal bipartite graphs.Polynomial time algorithms:

[Steiner-Stewart 87] Bipartite permutation graphs.[Brandstadt 89] Biconvex graphs.[Dahlhaus 94] Convex graphs.OPEN: 2D-graphs (or permutation graphs).

BipartitePermutation

⊂ Biconvex ⊂ Convex ⊂ 2DORG ⊂ ChordalBipartite⊂

2D-graphs


(cont.) Jump number

Properties of j(P)

[Habib 84] Comparability invariant: j(G).[Muller 90] NP-hard for chordal bipartite graphs.

Polynomial time algorithms:[Steiner-Stewart 87] Bipartite permutation graphs.[Brandstadt 89] Biconvex graphs.[Dahlhaus 94] Convex graphs.OPEN: 2D-graphs (or permutation graphs).



2D-graphs


(cont.) Jump number

Properties of j(P)

[Habib 84] Comparability invariant: j(G).[Muller 90] NP-hard for chordal bipartite graphs.Polynomial time algorithms:

[Steiner-Stewart 87] Bipartite permutation graphs.[Brandstadt 89] Biconvex graphs.[Dahlhaus 94] Convex graphs.OPEN: 2D-graphs (or permutation graphs).



2D-graphs


Permutation Graphs (2D-graphs)

Definition (2D-graphs)

Given a set V of points in the plane.G(V) is the graph where

ab is an edge if ax ≤ bx and ay ≤ by .


Two-Directional Orthogonal Ray Graphs (2DORG)

Definition (Bicolored 2D-graphs or 2DORG)

Given two sets A and B of points in the plane.G(A,B) is the bipartite graph on A ∪ B where

ab is an edge if a ∈ A, b ∈ B, ax ≤ bx and ay ≤ by .

Geometricformulation of a

2DORG as arectangle family R.


Outline



3 Main results

Cross-free Matchings and Biclique Covers.

Cross-free matchings

Edges ab and a′b′ cross if ab′ and a′b are also edges.

α∗(G) = maximum size of a cross-free matching.

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Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.

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Biclique Cover

Biclique = bipartite complete subgraph.

κ∗(G) = minimum size of a biclique-cover.




Edges ab and a′b′ cross if ab′ and a′b are also edges.α∗(G) = maximum size of a cross-free matching.

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d e f

Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.

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Biclique Cover







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d e f Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.

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Biclique Cover







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Biclique Cover

Biclique = bipartite complete subgraph.κ∗(G) = minimum size of a biclique-cover.





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Biclique Cover

Biclique = bipartite complete subgraph.κ∗(G) = minimum size of a biclique-cover.

α∗(G) ≤ κ∗(G).


α∗(G(A,B)) and κ∗(G(A,B)) in 2DORGs

Crossing edges = Overlapping rectangles

Maximal bicliques = Rectangle hitting sets

Proposition [ST11]: In a 2DORG with rectangles Rα∗(G(A,B)) = maximum independent set of R [MIS(R)].κ∗(G(A,B)) = minimum hitting set of R [MHS(R)].


α∗(G(A,B)) and κ∗(G(A,B)) in 2DORGs

Crossing edges = Overlapping rectangles

Maximal bicliques = Rectangle hitting sets

Can replace Rby the

inclusionwiseminimal

rectangles R↓.

Proposition [ST11]: In a 2DORG with rectangles Rα∗(G(A,B)) = maximum independent set of R [MIS(R)].κ∗(G(A,B)) = minimum hitting set of R [MHS(R)].


Outline



3 Main results

Linear Program Formulation for MIS(R↓)

z∗ = max∑

R∈R↓

xR

P =

∑R3q

xR ≤ 1, q ∈ Grid.

xR ≥ 0, R ∈ R↓.

1

Nonintegral Polytope.

1

-area

Integral

Theorem 1 [ST11]: In a 2DORG with minimal rectangles R↓The fractional solution with minimum weighted area is integral, i.e.:

arg min{ ∑

R∈R↓

area(R)xR : 1T x = z∗, x ∈ P}

is integral.

Proof: Uncrossing argument.


Min-Max relation

Theorem 2 [ST11]: For every 2DORG,

max. indep. set(R↓) = min. hitting set(R↓).α∗(G(A,B)) = κ∗(G(A,B)).

And we can compute them in O(n2.38)-time.

Proof has elements from [Frank 99] and [Benczur-Foster-Kiraly 99].


(sketch) Theorem 2: α∗ = κ∗

H : intersection graph of R↓.α∗ = MIS(R↓) = stability number of H.κ∗ = MHS(R↓) = clique covering number of H.

The only intersections in R↓ are:

Corner-free intersections or Corner intersections.

Perfect Case:

If R↓ only has corner-free-intersections, then H is a (perfect)comparability graph: R � S ⇐⇒ (Rx ⊇ Sx) and (Ry ⊆ Sy ).


(sketch) Theorem 2: α∗ = κ∗

H : intersection graph of R↓.α∗ = MIS(R↓) = stability number of H.κ∗ = MHS(R↓) = clique covering number of H.

The only intersections in R↓ are:

Corner-free intersections or Corner intersections.

Perfect Case:

If R↓ only has corner-free-intersections, then H is a (perfect)comparability graph: R � S ⇐⇒ (Rx ⊇ Sx) and (Ry ⊆ Sy ).

Therefore α∗ = κ∗ and we can compute them efficiently.


(cont.) Theorem 2: α∗ = κ∗

General Case: Build indep. set and hitting set in R↓ of same size .

1 Construct K ⊆ R↓ by greedily including rectangles in K notforming corner-intersection.

2 Since K is a corner-free-intersection familyMHS(K)=MIS(K)≤MIS(R↓)≤MHS(R↓)

= MHS(K)

3 Compute P, a minimum hitting set of K (with points in the Grid).

Swapping procedure.If p,q in P, with px < qx and py < qy s.t.

P′ = P \ {p,q} ∪ {(px ,qy ), (py ,qx)}is a hitting set for K then set P← P′.

We can show that final P is also a hitting set for R↓.






= MHS(K)3 Compute P, a minimum hitting set of K (with points in the Grid).









= MHS(K)










= MHS(K)




p

q






2 Since K is a corner-free-intersection familyMHS(K)=MIS(K)≤MIS(R↓)≤MHS(R↓) = MHS(K)




p

q



Conclusions and Other Results.

Algorithmic Results

O(n2.38) algorithm for the jump number, maximum cross-freematching and minimum biclique cover in any 2DORG.

Jump number in convex graphs: O(n2) algorithmimproving over O(n9) [Dahlhaus 94].Maximum weight cross-free matching is NP-complete in 2DORGs.O(n3) algorithm for weighted problem in biconvex and convexgraphs.



Algorithmic Results

O(n2.38) algorithm for the jump number, maximum cross-freematching and minimum biclique cover in any 2DORG.Jump number in convex graphs: O(n2) algorithmimproving over O(n9) [Dahlhaus 94].

Maximum weight cross-free matching is NP-complete in 2DORGs.O(n3) algorithm for weighted problem in biconvex and convexgraphs.



Algorithmic Results

O(n2.38) algorithm for the jump number, maximum cross-freematching and minimum biclique cover in any 2DORG.Jump number in convex graphs: O(n2) algorithmimproving over O(n9) [Dahlhaus 94].Maximum weight cross-free matching is NP-complete in 2DORGs.

O(n3) algorithm for weighted problem in biconvex and convexgraphs.



Algorithmic Results

O(n2.38) algorithm for the jump number, maximum cross-freematching and minimum biclique cover in any 2DORG.Jump number in convex graphs: O(n2) algorithmimproving over O(n9) [Dahlhaus 94].Maximum weight cross-free matching is NP-complete in 2DORGs.O(n3) algorithm for weighted problem in biconvex and convexgraphs.



DiscussionGeometric interpretation provides helpful intuition.

Max cross-free matching = Min biclique cover (in 2DORGs).This encompasses

Max antirectangle = Min rectangle cover (for biconvex boards)[Chaiken et al. 81].Max irredundant interval family = Min interval basis[Gyori 84, Frank 99].Max independent set = Min hitting set (in 2DORGs).

Part of our result can be seen as a non-trivial application of[Frank-Jordan 95].

OPENJump number of 2D-graphs.Approximation algorithms?



DiscussionGeometric interpretation provides helpful intuition.Max cross-free matching = Min biclique cover (in 2DORGs).This encompasses

Max antirectangle = Min rectangle cover (for biconvex boards)[Chaiken et al. 81].Max irredundant interval family = Min interval basis[Gyori 84, Frank 99].Max independent set = Min hitting set (in 2DORGs).

Part of our result can be seen as a non-trivial application of[Frank-Jordan 95].

OPENJump number of 2D-graphs.Approximation algorithms?


Thank you!


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