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Efficient Algorithms for the Zarankiewicz Problem

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Efficient Algorithms for the Zarankiewicz Problem Efficient algorithms for the Zarankiewicz problem Andrew Kay∗ School of Computing and Digital Technology, Birmingham City University April 19, 2016 Abstract The Zarankiewicz problem asks for the maximum number of 1s in an m×n matrix with no s×t minor containing only 1s. We present a general algorithm and a specific algorithm for the case s = t = 2, each substantially more efficient than previous work. The algorithms are based on a generalisation of Paige{Wexler canonical form for finite projective planes, and a new connection with symmetric inverse semigroups analogous to a connection between finite projective planes and symmetric groups. We obtain over 200 new exact values, and correct previously unreported errors in R. K. Guy's tables. Finally, we make some observations which may apply to the search for finite projective planes. Keywords. Zarankiewicz problem, rectangle-free, forbidden minor, finite projective plane, extremal combinatorics, computational combinatorics, constraint programming. 1 Introduction In 1951, Zarankiewicz posed some specific cases of a problem which, more generally, asks for the maximum number of 1s in an m×n matrix with no s×t minor containing only 1s. [Zar51] The problem is also stated in extremal graph theory as the maximal number of edges in a bipartite graph on vertex sets U ; V with jU j; jV j = (m; n) such that no s vertices from U and t vertices from V span a complete bipartite subgraph. [DHS13] The problem is difficult in general; exact values known to date are due to a few theorems establishing exact values for some infinite classes of parameters, a patchwork of upper and lower bounds none comprehensively sharper than another, ad hoc methods for particular cases, and more recently by computer search. Most effort has been focused on the case s = t = 2. The problem remains an active area of research; see e.g. [DHS13, DDR13, FS13, SP12,Wer12]. ∗Electronic address: [email protected] 1 1.1 Notation Matrices in this paper have entries in fI ; Og rather than f1; 0g. This unusual choice of notation is to reduce cognitive dissonance, as in Section 1.3 we will define an ordering with I < O. The notation also extends more naturally to matrices with entries in semigroups in Section 3.2. We interpret I as an incidence flag with no arithmetic properties. The set of natural numbers (including 0) is N, and N1 = N[f1g. The set of m×n matrices with entries in a set Σ is MatΣ(m; n). The weight w(A) of a matrix A 2 MatfI ;Og is its number of I entries. More generally, we allow a weight function w :Σ ! N to be extended to w : MatΣ(m; n) ! N via the formula m n X X X w(A) = w(Ai;j ) = jAjσw(σ) (1.1) i=1 j =1 σ2Σ where jAjσ is the number of σ entries in A. The specific case has w(A) = jAjI , implying w(I ) = 1 and w(O) = 0. Definition 1.2. An (s; t)-rectangle is an s×t minor containing only I entries. A matrix with no (s; t)-rectangles is (s; t)-rectangle-free. Definition 1.3. z(m; n; s; t) is the maximum weight of an (s; t)-rectangle-free m×n matrix.1 For brevity, in the case s = t = 2 we will simply write rectangle, rectangle-free and z(m; n). We will leave O entries blank where this aids readability. When referring to the rows and columns of a particular matrix, we will write Ri for row i and Cj for column j . n As usual, [n] = f1; 2;:::; ng and is the binomial symbol. < is lexicographic order. k lex It will sometimes be useful to identify matrices in MatfI ;Og(m; n) with subsets of [m] × [n]; in this case, (i; j ) 2 A if and only if Ai;j = I . We also write • dom A ⊆ [m] for the domain, and ran A ⊆ [n] for the range, • U 7! V for the set of partial functions from U to V , 2 • U 7 V for the set of partial injections from U to V , and • B ◦ A for backward relational composition, as in the Z notation. [Spi92] 1Earlier literature (e.g. [Guy69, KST54, Rom75]) refers instead to the minimum weight such that every m×n matrix has an (s; t)-rectangle. This is of course z(m; n; s; t) + 1. 2Sometimes called partial bijections, or partial permutations. 2 1.2 Bounds and exact values We summarise a number of theorems establishing bounds and exact values for z(m; n; s; t). Some theorems are asymmetric in m and n, in that they might give a sharper bound on the transpose matrix. Clearly z(m; n; s; t) = z(n; m; t; s). The first result is used in various arguments e.g. in [Guy69,KST54,Rom75]. Definition 1.4. For A 2 MatΣ(m; n) with a weight function w, m 1. The row weight distribution of A is a vector r 2 N with ri = w(Ri ). n 2. The column weight distribution of A is a vector c 2 N with cj = w(Cj ). Theorem 1.5. Call a vector r 2 Nm \admissible" if m X ri n (s − 1) (1.6) t 6 t i=1 Then every (s; t)-rectangle-free matrix has an admissible row weight distribution, and m X z(m; n; s; t) max ri (1.7) 6 r i=1 where the maximum is taken over all admissible vectors. Since the sum in (1.6) is a convex function of r, the maximum in (1.7) can be attained by a vector with every jri − rk j 6 1. Therefore, this bound can be calculated directly without iterating over admissible vectors. Note that an admissible vector is not necessarily realisable as the row weight distribution of an (s; t)-rectangle-free matrix. Proof. The sum in (1.6) counts the total number of combinations of t columns spanned by the I entries in each row of A. If the row weight distribution r of A is not admissible, then by the pigeonhole principle there is a combination of t columns such that A has at least s rows with I entries in those columns. Hence, A is not (s; t)-rectangle-free. Therefore an extremal (s; t)-rectangle-free matrix A has an admissible row weight distribu- Pm tion. w(A) = i=1 ri gives the result. The next theorem achieves equality for m n, using rows of weight t and t − 1. n Theorem 1.8 (Culikˇ [Cul56])ˇ . If m (s − 1) , then > t n z(m; n; s; t) = (t − 1)m + (s − 1) t 3 The next theorem achieves equality in the case s = t = 2 for some square matrices. 2 Theorem 1.9 (Reiman [Rei58]). Let n = k + k + 1 for k > 2. Then 2 z(n; n) 6 (k + k + 1)(k + 1) and a rectangle-free matrix of this weight must be the incidence matrix of a finite projective plane of order k. The only known orders k are the prime powers. The next theorem also uses finite projective planes, and extends a lower bound of Dybizba´nski,Dzido & Radziszowski [DDR13]. 2 Theorem 1.10 (Deletion principle). Let n = k + k + 1 where k > 2 is the order of a finite projective plane. Then for c; d 2 N, then 2 z(n − c; n − d) > (k + k + 1 − c − d)(k + 1) + Xk (c; d) (1.11) where Xk (c; d) is the maximum weight of a c×d minor of an incidence matrix of a finite projective plane of order k. We have Xk (0; d) = 0 Xk (1; d) = minfd; k + 1g Xk (2; d) = minfd + 1; 2k + 2g for d > 1 Xk (3; d) = minfd + 3; 3k + 3g for d > 3 Xk (4; d) = minfd + 6; 4k + 4g for d > 6 For k a power of 2,Xk (7; 7) = 21. Values valid for all k are given in the table below. c; d 1 2 3 4 5 6 7 1 1∗ 2 3 2 3∗ 4 5 6 3 6∗ 7 8 9 4 9 10 12 5 12 14 15 6 16 18 Furthermore, (1.11) is an equality in the cases marked ∗. Proof. We obtain a rectangle-free matrix satisfying the lower bound by deleting c rows and d columns from the incidence matrix of a finite projective plane of order k. The weight of the resulting matrix is given by (1.11) when Xk (c; d) is the weight of the c×d minor of their intersection, by the inclusion/exclusion principle. For Xk (1; d) choose any row and any minfd; k + 1g columns intersecting that row. For Xk (2; d) with d > 1, choose any two rows, the column intersecting both, and any other minfd − 1; 2kg columns intersecting either. 4 In the remaining cases, by the non-degeneracy axiom of a finite projective plane, there are four columns C1;:::; C4 such that no row intersects any three. There must be six distinct rows R1;:::; R6 intersecting each pair; then the three pairs of rows (R1; R6), (R2; R5) and (R3; R4) must each intersect in distinct columns C5; C6; C7. C1 C2 C3 C4 C5 C6 C7 R1 0 II I 1 R2 B II I C B C R3 B II I C (1.12) B C R4 B II I C R5 @ II I A R6 II I It follows that any incidence matrix of a finite projective plane has (1.12) as a minor, and therefore has c×d (or d×c) minors of weight Xk (c; d) as given in the table above.
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