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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 2 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 3 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 4 / 22
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.”
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 5 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 6 / 22
Known results and conjectures
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 ).
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 7 / 22
Known results and conjectures
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 8 / 22
Known results and conjectures
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 9 / 22
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}.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 10 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 11 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 12 / 22
Maxcut and the Goemans-Williamson bound
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 13 / 22
Maxcut and the Goemans-Williamson bound
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)
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 14 / 22
Maxcut and the Goemans-Williamson bound
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 15 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 16 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 17 / 22
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}).
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 18 / 22
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).
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 19 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 20 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 21 / 22
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.
Etienne de Klerk (Tilburg Uni.) Improved bounds on crossing numbers of graphs 22 / 22