A LOWER BOUND ON THE LENGTH OF BASIC MINIMAL 1-( 3 t + 1 , 3 )

In a previous paper the author found some minimal 1(v,3) designs with large b, by exhaustive search. Here some further such are found, by more ad-hoc methods. A construction is given for v = 3t + 1 for t ≥ 2, where b grows quadratically with v. AMS Subj. Classification: 05B05


Introduction
Suppose v and k are integers with 1 ≤ k ≤ v.A 1-design with parameters v and k is a v row by b column matrix for some b, with elements either 0 or 1, such that each column contains k 1's, and each row contains r 1's for some r.Such a matrix is exactly the incidence matrix of a "biregular bipartite graph", with v vertices of degree r in one class, and b vertices of degree k in the other class.The notation "1-(v,k) design will be used to denote such designs.
There is a well-known system of linear equations associated with 1-designs.Various facts of interest about this system are proved in [1].The following further fact is of interest, both for the theory of 1-designs and for computations.Proof.Suppose first that S is nondegenerate, i.e., |S| = v where there are v rows.Let M be the matrix [C 1 , . . ., C v ] where S = {C 1 , . . ., C v }.Let x be the column vector of indeterminates x C 1 , . . ., x Cv .The system M x = J v×1 (see [1] for notation) has a unique solution x.D is obtained when x is converted to an integer vector with the integers having no common divisor.In general, suppose T 1 and T 2 are extensions of S to bases.It is readily seen that in the primal simplex tableau there is a sequence of level 0 pivots from T 1 to T 2 .Further, the solutions for T 1 and T 2 have the same restrictions to {x C : C ∈ S}.See [5] for terminology.

Long Designs
Theorem 6 and 7 of [1] give bounds on the length b of minimal and basic minimal 1-(v,k) designs.It is of interest to both design theory and to linear programming theory to obtain improvements to these bounds, and to obtain lower bounds.As seen in [1], these questions are of interest even for k = 2.
For 5 ≤ v ≤ 8 the exact bound for basic minimal 1-designs was determined in [1] by exhaustive search using vertex enumeration.For k = 3 these values are b=5, b=21, and b=48 respectively.(There is an error in Table 1 of [1]; the number of non-isomorphic basic minimal 1-(6,3) designs is 3, not 4).
In this paper, a lower bounds for basic minimal solutions with k = 3 will be given.Before giving this, a method for obtaining examples and lower bounds for small v will be given, which uses random search rather than exhaustive search.
Using notation as in [1], let M be the matrix with v rows labeled 0, . . ., v −1 and v k columns, the k element subsets of {0, . . ., v − 1}.Let M + H be the matrix derived from M by subtracting row 0 from the other rows and replacing it by a row of all 1's.A vector x is a solution to M x = J v×1 iff y = (v/k)x is a solution to M + H y = e 0 where e s denotes the (column) unit vector with 1 in row s.It follows that the null spaces of M and M + H are the same, and so the sets of columns which are bases is the same (labeling a column of M + H with the column of M from which it is derived).
Solutions to the LP M + H x = e 0 , x C ≥ 0 for all C will be considered.This LP is in standard form, and is readily solved using the primal simplex method, as described in [5] for example.The two-stage method for finding an initial feasible basis may be avoided, if a feasible basis is known.The columns of it may be transformed into distinct unit vectors by an arbitrary sequence of pivots on the columns of the basis.
For gcd(v, k) = 1 define the cyclic design to be that where column i is The theorem follows by circulant matrix theory (see [3]).
Theorem 3. Suppose gcd(v, k) = 1 and v = qk + s where q ≥ 2 and 0 < s < k.Let x be the vector where x C = k/v if C has 1's in rows kt to kt + k − 1 for some t with 0 ≤ t ≤ q − 2; x C = 1/v if C is one of the columns of the cyclic design in rows k(t − 1) to v − 1; and x C = 0 otherwise.Then x is a solution to M + H = e 0 which maximizes x {0,...,k−1} .
Proof.Let B be the matrix where for j < v − k column j has 1's in rows j, . . ., j + k − 1; and in the remaining columns there is a copy of the cyclic design in the lower right.B is block upper triangular, where the blocks along the diagonal are invertible (using theorem 2), so B is basic.Direct computation shows that Bx = (k/v)J v×1 .Direct computation also shows that the top row of (B + H ) −1 has k/v in column 0 and −1/v in the other columns.Let c be the (row) cost vector, with −1 in column 0 and 0's elsewhere.Let c B be the similarly defined vector of length v.As noted in Section 2.6 of [5], the relative cost vector of the simplex tableau at the basis B equals c − c B (B + H ) −1 M + H . Direct computation shows that this is 1 in column C if 0 ∈ C, except for column 0, which is 0; and 0 in the remaining columns.
The maximum value of x C for the cyclic design is 1/v.Starting from this basis, the simplex algorithm may be run with random pivots.The basic feasible solution with the largest value of b can be monitored.For k = 3, this was done for 7 ≤ v ≤ 20 with gcd(v, 3) = 1.For a given v, 100,000 repetitions were performed.In Figure 1, the value of b/v 3 is plotted; b seems to be growing faster than cubically.

Lower Bound on b
The computations of [1] show that there is a single basic minimal 1-(7,3) design of length 21; Figure 2 shows the columns, with their multiplicities.This has been arranged so that a pattern may be seen, which may be generalized (some designs of length 10 were also examined).

Bounding x max
Solutions with x {0,...,k−1} = k/v are clearly of little interest.This suggests modifying the LP by adding constraints x C + y C = x max , y C ≥ 0. Basic feasible solutions of M + H x = e 0 , with integer value having a greatest common divisor of 1, are minimal; this is a fact of LP theory, tacitly assumed in [1].With slack variables added as just indicated, the restriction of a basic solution to the x C may no longer be basic in the original LP.
The cost of the slack variables must be specified.In ordinary use this is 0 (see Section 3-2 of [2]).For this paper, this value is adopted.
By results of [1], for a minimal solution which is not basic, b ≤ 588.The modified LP was run with values of x max = n/d, where 2 ≤ d ≤ 84 or d ≤ 588 and d mod 7 = 0, 1/7 ≤ n/d ≤ 3/7, and gcd(n, d) = 1 (there are 4333 such n/d).
Each LP has a solution where the cost is x max ; further b equals d if d mod 7 = 0, else 7d.Using the 33395 basic minimal solutions (see [1]), it may be determined that for each modified LP, the optimum solution found by the simplex method is an integral linear combination of basic minimal solutions.In the cases where x max is the cost of a basic minimal solution, the optimal solution is basic minimal.These facts suggest that for v = 7, k = 3, minimal solutions are basic minimal; further remarks are omitted here.
Received: May 13, 2014 c 2014 Academic Publications, Ltd. url: www.acadpubl.euTheorem 1. Suppose D is a basic minimal 1-design, and S is the set of its columns.Then D is determined by S.