Improved bounds on crossing numbers of graphs via ...rdejong/diamantslides/deklerk.pdf · Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 2 / 22. De

Improved bounds on crossing numbers of graphs viasemidefinite programming

Etienne de Klerk‡ and Dima Pasechnik

‡Tilburg University, The Netherlands

DIAMANT symposium, November 2011

Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 1 / 22

Outline

Outline

The (two page) crossing numbers of complete and complete bipartite graphs.

A maximum cut formulation for the two page crossing number of Kn.

The Goemans-Williamson maximum cut bound and its implications.

A nonconvex quadratic programming relaxation of the two page crossingnumber of Km,n.

A semidefinite programming relaxation of the quadratic program and itsimplications.


Definitions

Crossing number of a graph

Definition

The crossing number cr(G ) of a graph G = (V ,E ) is the minimum number ofedge crossings that can be achieved in a drawing of G in the plane.

Example: the complete bipartite graph

An optimal drawing of K4,5 with cr(K4,5) = 8 edge crossings.


Definitions

Two-page crossing number of a graph

Definition

In a two-page drawing of G = (V ,E ) all vertices V must be drawn on a straightline (resp. circle) and all edges either above/below the line (resp. inside/outsidethe circle). The two-page crossing number ν2(G ) corresponds to two-pagedrawings of G .

Example: the complete graph K5

Equivalent two-page drawings of K5 with ν2(K5) = 1 crossing.


Applications and hardness results

Applications and complexity

Crossing numbers are of interest for graph visualization, VLSI design,quantum dot cellular automata, ...

It is NP-hard to compute cr(G ) or ν2(G ) [Garey-Johnson (1982), Masuda et al.

(1987)];

The (two-page) crossing numbers of Kn and Kn,m are not known, ...

Crossing number of Kn,m known as Turan brickyard problem — posed byPaul Turan in the 1940’s.

Erdos and Guy (1973):

”Almost all questions that one can ask about crossing numbers remain unsolved.”


Known results and conjectures

The Zarankiewicz conjecture

Km,n can be drawn in the plane with at most Z (m, n) edges crossing, where

Z (m, n) =

⌊m − 1

2

⌋⌊m

2

⌋⌊n − 1

2

⌋⌊n

2

⌋.

A drawing of K4,5 with Z(4, 5) = 8 crossings.

Zarankiewicz conjecture (1954)

cr(Km,n)?= Z (m, n).

Known to be true for min{m, n} ≤ 6 (Kleitman, 1970), and some special cases.



The 2-page Zarankiewicz conjecture

The Zarankiewicz drawing may be mapped to a 2-page drawing:

”Straighten the dotted line”.

2-page Zarankiewicz conjecture

ν2(Km,n)?= Z (m, n).

Weaker conjecture since cr(G ) ≤ ν2(G ).



The (2-page) Harary-Hill conjecture

Conjecture (Harary-Hill (1963))

cr(Kn)?= ν2(Kn)

?= Z (n) :=

1

4

⌊n

2

⌋⌊n − 1

2

⌋⌊n − 2

2

⌋⌊n − 3

2

⌋NB: it is only known that cr(Kn) ≤ ν2(Kn) ≤ Z (n) in general.

Example: the complete graph K5

Optimal two-page drawings of K5 with Z (5) = 1 crossing.



Some known results

Theorem (De Klerk, Pasechnik, Schrijver (2007))

One has

1 ≥ limn→∞

cr(Kn)

Z (n)≥ 0.8594, 1 ≥ lim

n→∞

cr(Km,n)

Z (m, n)≥ 0.8594 if m ≥ 9,

Theorem (Pan and Richter (2007), Buchheim and Zheng (2007))

cr(Kn) = Z (n) if n ≤ 12, ν2(Kn) = Z (n) if n ≤ 14.


New results

New results (this talk)

Theorem (De Klerk and Pasechnik (2011))

For the complete graph Kn, one has

1 ≥ limn→∞

ν2(Kn)

Z (n)≥ 0.9253

andν2(Kn) = Z (n) if n ≤ 18 or n ∈ {20, 22}.

For the complete bipartite graph Km,n, one has

limn→∞

ν2(Km,n)

Z (m, n)= 1 if m ∈ {7, 8}.


New results

New results: outline of the proofs

For Kn:

The problem of computing ν2(Kn) has a formulation as a maximum cutproblem (Buchheim and Zheng (2007));

The new results for ν2(Kn) follow by computing the Goemans-Williamsonmaximum cut bound for n = 899.

The Goemans-Williamson bound is computed using semidefiniteprogramming (SDP) software and using algebraic symmetry reduction.

For Km,n:

We will formulate a (nonconvex) quadratic programming (QP) lower boundon ν2(Km,n).

Subsequently we compute an SDP lower bound on the QP bound for m = 7,again using algebraic symmetry reduction.


Maxcut and the Goemans-Williamson bound

The maximum cut problem for graphs

Definition

For G = (V ,E ) and a subset W ⊆ V , cutW (G ) denotes the set of edges withprecisely one endpoint in W , and is called the edges in the cut W . The weight ofa maximum cut is

maxcut(G ) := maxW⊆V

|cutW (G )|.

Example:

The black vertices yield the maximum cut in the figure, maxcut(G ) = 5.



ν2(Kn) as a maximum cut problem

Results of Buchheim and Zheng (2007) imply that ν2(Kn) may be obtained bysolving a maximum cut problem in a suitable graph, Gn = (Vn,En), say.

Vn is the set of chords in the standard n-cycle, thus |Vn| =(n2

)− n;

Two vertices are adjacent if the corresponding chords cross when drawninside the cycle, thus |En| =

(n4

).

Illustration:

v1

v2v3

v4v5

The chords {v1, v4} and {v2, v5} form adjacent vertices in the graph G5.



ν2(Kn) as a maximum cut problem (ctd)

Lemma (cf. Buchheim and Zheng (2007))

One hasν2(Kn) = |En| −maxcut(Gn).

Proof sketch: Given a two page (circle) drawing of Kn, define a cut W ⊂ Vn asthe chords that are drawn inside the circle.

Using this formulation, Buchheim and Zheng (2007) showed thatν2(Kn) = Z (n) for n ≤ 14, by using a branch and bound method.

Using the maximum cut software BiqMac (Rendl, Rinaldi, and Wiegele(2010)) we could extend this result for n up to 18, and for n = 20, 22.

Further progress possible by considering the Goemans-Williamson maximum cutbound ... (next slide)



The Goemans-Williamson maxcut bound

Goemans-Williamson bound (dual formulation)

For a graph G = (V ,E ) with Laplacian matrix L define:

GW(G ) := miny∈R|V |

{∑i∈V

yi

∣∣∣∣ Diag(y)− 1

4L � 0

},

where

A � 0 means the matrix A is hermitian positive semidefinite;

Diag(y) is the diagonal matrix with the vector y as diagonal.

Theorem (Goemans-Williamson (1995))

0.878GW(G ) ≤ maxcut(G ) ≤ GW(G ).

Since the Laplacian of Gn has block-circulant structure, we may compute GW(Gn)for n up to 1000 via symmetry reduction.


New result for ν2(Kn)

Computational results with SDPT3 solver

600100 500400300200

0.922

0.925

0.924

0.926

0.927

0.923

700 800 900

The ratio(n

4)−GW(Gn)

Z(n) for n = 99, 199, . . . , 899.


New result for ν2(Kn)

Implication for general n

Lemma

For any integer m > 3,

limn→∞

ν2(Kn)

Z (n)≥ 64

m(m − 1)(m − 2)(m − 3)ν2(Km).

As a consequence, setting m = 899 and using

ν2(Km) ≥(m

4

)− GW(Gm)

we obtain:

Corollary

limn→∞

ν2(Kn)/Z (n) ≥ 0.9253.


Results for Km,n

Drawings of Km,n

Consider a drawing of Km,n with the n coclique colored red, and the m cocliqueblue.

Definition

Each red vertex r has a position p(r) ∈ {1, . . . ,m} in the drawing, and a set ofincident edges U(r) ⊆ {1, . . . ,m} drawn in the upper half plane. We say r is ofthe type (p(r),U(r)). The set of all possible types is denoted by Types(m), i.e.|Types(m)| = m2m.

b3b2b1 r b4 b5xr ′

In the figure, r has type (p(r),U(r)) = (2, {1, 2, 3, 5}).


Results for Km,n

A quadratic programming relaxation of ν2(Km,n)

We define a m2m ×m2m matrix Q with rows/colums indexed by Types(m).

Definition

Let σ, τ ∈ Types(m). Define Qτ,σ as the number of unavoidable edge crossings ina 2-page drawing of K2,m, where the vertices from the 2-coclique have type σ andτ respectively in the drawing.

Lemma

ν2(Km,n) ≥ n2

2

(minx∈∆

xTQx

)− m(m − 1)n

4

where ∆ =

{x ∈ Rm2m

∣∣∣∣ ∑i xi = 1, xi ≥ 0

}is the standard simplex.

This is a nonconvex quadratic program — we again use a semidefiniteprogramming relaxation (next slide).


Results for Km,n

A semidefinite programming relaxation of ν2(Km,n) (ctd)

Standard semidefinite programming relaxation:

minx∈∆

xTQx ≥ min{

trace(QX )∣∣ trace(JX ) = 1, X � 0, X ≥ 0

},

where J is the all-ones matrix and X ≥ 0 means X is entrywise nonnegative.

We may again perform symmetry reduction using the structure of Q ...

... namely Q is a block matrix with 2m × 2m circulant blocks (afterreordering rows/columns).

The reduced problem has 2m linear matrix inequalities involving(2m−1)× (2m−1) matrices.


Results for Km,n

Computational results and implications

We could compute the SDP bound for m = 7 to obtain

ν2(K7,n) ≥ (9/4)n2 − (21/2)n = Z (7, n)− O(n).

Since ν2(K8,n) ≥ 8ν2(K7,n)/6, we also get ν2(K8,n) ≥ 3n2− 14n = Z (8, n)−O(n).

Corollary

limn→∞

ν2(Km,n)/Z (m, n) = 1 for m = 7 and 8.

In words, the 2-page Zarankiewicz conjecture is true asymptotically for m = 7 and8.


Results for Km,n

Conclusion and summary

We demonstrated improved asymptotic lower bounds on ν2(Kn), ν2(K7,n),and ν2(K8,n).

The proofs were computer-assisted, and the main tools were semidefiniteprogramming (SDP) relaxations and symmetry reduction.

The SDP relaxation was too large to solve for ν2(K9,n) — challenge for SDPcommunity.

Preprint available at Optimization Online and arXiv.


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Improved bounds on crossing numbers of graphs via ...rdejong/diamantslides/deklerk.pdf · Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 2 / 22. De