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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
Jump Number Problem
a b c
d e f
P
a
b
c
f
d
e
4 Jumps
a b c
d e f
P
a
b
d
c
f
e
a
b
c
f
d
e
4 Jumps 3 Jumps
a b c
d e f
P
Jump number of a poset PFind a linear extension (total order)with minimum number of jumps j(P).
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 3
Jump Number Problem
a b c
d e f
P
a
b
c
f
d
e
4 Jumps
a b c
d e f
P
a
b
d
c
f
e
a
b
c
f
d
e
4 Jumps 3 Jumps
a b c
d e f
P
Jump number of a poset PFind a linear extension (total order)with minimum number of jumps j(P).
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 3
Jump Number Problem
a b c
d e f
P
a
b
c
f
d
e
4 Jumps
a b c
d e f
P
a
b
d
c
f
e
a
b
c
f
d
e
4 Jumps 3 Jumps
a b c
d e f
P
Jump number of a poset PFind a linear extension (total order)with minimum number of jumps j(P).
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 3
(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
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 4
(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
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 4
(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
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 4
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 .
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 5
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 6
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 6
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 6
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.
a b c
d e f
Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.
a
b
d
c
f
e
Biclique Cover
Biclique = bipartite complete subgraph.
κ∗(G) = minimum size of a biclique-cover.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 7
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.
a b c
d e f
Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.
a
b
d
c
f
e
Biclique Cover
Biclique = bipartite complete subgraph.
κ∗(G) = minimum size of a biclique-cover.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 7
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.
a b c
d e f Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.
a
b
d
c
f
e
Biclique Cover
Biclique = bipartite complete subgraph.
κ∗(G) = minimum size of a biclique-cover.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 7
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.
a b c
d e f Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.
a
b
d
c
f
e
Biclique Cover
Biclique = bipartite complete subgraph.
κ∗(G) = minimum size of a biclique-cover.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 7
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.
a b c
d e f Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.
a
b
d
c
f
e
Biclique Cover
Biclique = bipartite complete subgraph.κ∗(G) = minimum size of a biclique-cover.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 7
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.
a b c
d e f Fact [Muller 90]:For G chordal bipartite.α∗(G) + j(G) = n − 1.
a
b
d
c
f
e
Biclique Cover
Biclique = bipartite complete subgraph.κ∗(G) = minimum size of a biclique-cover.
α∗(G) ≤ κ∗(G).
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 7
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)].
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 8
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)].
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 8
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)].
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 8
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 9
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 9
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].
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 10
(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 ).
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 11
(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 ).
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 11
(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 ).
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 11
(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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 11
(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↓.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 12
(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↓.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 12
(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↓.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 12
(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′.
p
q
We can show that final P is also a hitting set for R↓.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 12
(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′.
p
q
We can show that final P is also a hitting set for R↓.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 12
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 13
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 13
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 13
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.
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 13
Conclusions and Other Results.
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?
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 14
Conclusions and Other Results.
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?
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 14
Conclusions and Other Results.
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?
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 14
Conclusions and Other Results.
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?
Soto, Telha - MIT Jump Number of 2DORGS IPCO 2011 14
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