Showing papers in "SIAM Journal on Discrete Mathematics in 2003"
TL;DR: An upper bound on the contrast of any (3,n)-threshold VCS is proved and a scheme in the described family has optimal contrast and is completely characterize a contrast optimal (n-1,n) VCS in canonical form.
Abstract: A (k,n)-threshold visual cryptography scheme (VCS) is a method to encode a secret image SI into n shadow images called shares such that any k or more shares enable the "visual" recovery of the secret image. However, by inspecting less than k shares one cannot gain any information on the secret image. The "visual" recovery consists of copying the shares onto transparencies and then stacking them. Any k shares will reveal the secret image without any cryptographic computation. In this paper we analyze the contrast of the reconstructed image for a (k,n)-threshold VCS. We define a canonical form for a (k,n)-threshold VCS and provide a characterization of a (k,,n)-threshold VCS. We completely characterize a contrast optimal (n-1,n)-threshold VCS in canonical form. Moreover, for $n\geq 4$, we provide a contrast optimal (3,n)-threshold VCS in canonical form. We first describe a family of (3,n)-threshold VCS achieving various values of contrast and pixel expansion. Then we prove an upper bound on the contrast of any (3,n)-threshold VCS and show that a scheme in the described family has optimal contrast. Finally, for k=4,5 we present two schemes with contrast asymptotically equal to 1/64 and 1/256, respectively.
186 citations
TL;DR: In this paper, it was shown that the chromatic number of a power graph Gk of a planar graph G is at most Ω(n 9\Delta /5 \rceil), where n is the maximum degree of the graph.
Abstract: We give nontrivial bounds for the inductiveness or degeneracy of power graphs Gk of a planar graph G. This implies bounds for the chromatic number as well, since the inductiveness naturally relates to a greedy algorithm for vertex-coloring the given graph. The inductiveness moreover yields bounds for the choosability of the graph. We show that the inductiveness of a square of a planar graph G is at most $\lceil 9\Delta /5 \rceil$, for the maximum degree $\Delta$ sufficiently large, and that it is sharp. In general, we show for a fixed integer $k\geq1$ the inductiveness, the chromatic number, and the choosability of Gk to be $O(\Delta^{\lfloor k/2 \rfloor})$, which is tight.
153 citations
TL;DR: In this article, the problem of determining whether a given function belongs to a certain class of Boolean functions or not is considered. But the problem is not restricted to monomials, but also to monotone disjunctive normal forms.
Abstract: We consider the problem of determining whether a given function $f:{\{0,1\}}^n\to{\{0,1\}}$ belongs to a certain class of Boolean functions $\cal F$ or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter $\epsilon$, we would like to decide whether $f \in \cal F$ or whether it differs from every $g\in \cal F$ on more than an $\epsilon$-fraction of the domain elements. The classes of functions we consider are singleton ("dictatorship") functions, monomials, and monotone disjunctive normal form functions with a bounded number of terms. In all cases we provide algorithms whose query complexity is independent of n (the number of function variables), and linear in $1/\epsilon$.
121 citations
TL;DR: All balanced list channel assignment problems (G,L,w) which admit a proper coloring are characterized, which means that each graph with maximum degree $\Delta\ge 2$ has an L(2,1)-labeling using integers $0,\ldots,\Delta^2+\Delta-1$.
Abstract: A list channel assignment problem is a triple (G,L,w), where G is a graph, L is a function which assigns to each vertex of G a list of integers (colors), and w is a function which assigns to each edge of G a positive integer (its weight). A coloring c of the vertices of G is proper if c(v)\in L(v)$ for each vertex v and $|c(u)-c(v)|\ge w(uv)$ for each edge uv. A weighted degree $\deg_w(v)$ of a vertex v is the sum of the weights of the edges incident with v. If G is connected, $|L(v)|>\deg_w(v)$ for at least one v, and $|L(v)|\ge\deg_w(v)$ for all v, then a proper coloring always exists. A list channel assignment problem is balanced if $|L(v)|=\deg_w(v)$ for all v. We characterize all balanced list channel assignment problems (G,L,w) which admit a proper coloring. An application of this result is that each graph with maximum degree $\Delta\ge 2$ has an L(2,1)-labeling using integers $0,\ldots,\Delta^2+\Delta-1$.
115 citations
TL;DR: It turns out that $\gamma$ is a useful cousin of the expected hitting time of a Markov chain but is defined, for example, even for random walks on infinite graphs.
Abstract: We analyze and solve a game in which a player chooses which of several Markov chains to advance, with the object of minimizing the expected time (or cost) for one of the chains to reach a target state. The solution entails computing (in polynomial time) a function $\gamma$---a variety of "Gittins index"---on the states of the individual chains, the minimization of which produces an optimal strategy.
It turns out that $\gamma$ is a useful cousin of the expected hitting time of a Markov chain but is defined, for example, even for random walks on infinite graphs. We derive the basic properties of $\gamma$ and consider its values in some natural situations.
71 citations
TL;DR: Algorithms are presented that construct an implicit representation of clusterings of all but an $\epsilon$-fraction of the points in X that have optimal (or close to optimal) cost and are used to find approximately good clusterings.
Abstract: A set X of points in $\Re^d$ is (k,b)-clusterable if X can be partitioned into k subsets (clusters) so that the diameter (alternatively, the radius) of each cluster is at most b. We present algorithms that, by sampling from a set X, distinguish between the case that X is (k,b)-clusterable and the case that X is $\epsilon$-far from being (k,b')-clusterable for any given $0<\epsilon\leq 1$ and for $b' \geq b$. By $\epsilon$-far from being (k,b')-clusterable we mean that more than $\epsilon\cdot|X|$ points should be removed from X so that it becomes (k,b')-clusterable. We give algorithms for a variety of cost measures that use a sample of size independent of |X| and polynomial in k and $1/\epsilon$.
Our algorithms can also be used to find approximately good clusterings. Namely, these are clusterings of all but an $\epsilon$-fraction of the points in X that have optimal (or close to optimal) cost. The benefit of our algorithms is that they construct an implicit representation of such clusterings in time independent of |X|. That is, without actually having to partition all points in X, the implicit representation can be used to answer queries concerning the cluster to which any given point belongs.
61 citations
TL;DR: An exact formula is obtained for the probability that the Krylov space generated by a fixed linear mapping and a random set of elements in a vector space over a finite field equals the space itself.
Abstract: Motivated by a connection with block iterative methods for solving linear systems over finite fields, we consider the probability that the Krylov space generated by a fixed linear mapping and a random set of elements in a vector space over a finite field equals the space itself. We obtain an exact formula for this probability and from it we derive good lower bounds that approach 1 exponentially fast as the size of the set increases.
43 citations
TL;DR: A polynomial-time algorithm for the more general problem of simultaneous covering in the case when T is an arborescence is described and NP-completeness results for fairly specialized classes of the simultaneous cover problem are given.
Abstract: Damaschke, Muller, and Kratsch [Inform. Process. Lett., 36 (1990), pp. 231--236] gave a polynomial-time algorithm to solve the minimum dominating set problem in convex bipartite graphs $B=(X \cup Y,E)$, that is, where the nodes in Y can be ordered so that each node of X is adjacent to a contiguous sequence of nodes. Gamble et al. [Graphs Combin., 11 (1995), pp. 121--129] gave an extension of their algorithm to weighted dominating sets. We formulate the dominating set problem as that of finding a minimum weight subset of elements of a graphic matroid, which covers each fundamental circuit and fundamental cut with respect to some spanning tree T. When T is a directed path, this simultaneous covering problem coincides with the dominating set problem for the previously studied class of convex bipartite graphs. We describe a polynomial-time algorithm for the more general problem of simultaneous covering in the case when T is an arborescence. We also give NP-completeness results for fairly specialized classes of the simultaneous cover problem. These are based on connections between the domination and induced matching problems.
43 citations
TL;DR: This approximation scheme is very general and it can be extended to the case of job shop scheduling problems with release and delivery times, multistageJob shops, dag job shops, and preemptive variants of most of these problems.
Abstract: In this paper we present a linear time approximation scheme for the job shop scheduling problem with a fixed number of machines and fixed number of operations per job This improves on the previously best $2+\epsilon$, $\epsilon > 0$, approximation algorithm for the problem by Shmoys, Stein, and Wein [SIAM J Comput, 23 (1994), pp 617--632] Our approximation scheme is very general and it can be extended to the case of job shop scheduling problems with release and delivery times, multistage job shops, dag job shops, and preemptive variants of most of these problems
41 citations
TL;DR: For a CCC problem on n variables and m constraints, this problem is a generalization of the known maximum (or minimum) closure problem and the isotonic regression problem and an algorithm is described that has the complexity of the minimum cut problem plus the difficulty of finding the minima of up to n convex functions.
Abstract: Many applications in the area of production and statistical estimation are problems of convex optimization subject to ranking constraints that represent a given partial order. This problem, which we call the convex cost closure (CCC) problem, is a generalization of the known maximum (or minimum) closure problem and the isotonic regression problem. For a CCC problem on n variables and m constraints we describe an algorithm that has the complexity of the minimum cut problem plus the complexity of finding the minima of up to n convex functions. Since the CCC problem is a generalization of both minimum cut and minimization of n convex functions, this complexity is the fastest one possible. For the quadratic problem the complexity of our algorithm is strongly polynomial, $O(mn\log {\frac{n^2}{m}})$. For the isotonic regression problem the complexity is O(n log U,) for U the largest range for a variable value.
41 citations
TL;DR: A deterministic algorithm that constructs a graph of girth logk(n) + O(1) and minimum degree k-1, taking number of nodes n and number of edges e as input, which works for all values of n and $k < \frac {n}{3}$, unlike most of the earlier constructions.
Abstract: We give a deterministic algorithm that constructs a graph of girth logk(n) + O(1) and minimum degree k-1, taking number of nodes n and number of edges $e = {\left \lfloor nk / 2 \right \rfloor }$ (where $k < \frac {n}{3}$) as input. The degree of each node is guaranteed to be k-1, k, or k+1, where k is the average degree. Although constructions that achieve higher values of girth---up to $\frac {4}{3} \log_{k-1}{(n)}$---with the same number of edges are known, the proof of our construction uses only very simple counting arguments in comparison. Our method is very simple and perhaps the most intuitive: We start with an initially empty graph and keep introducing edges one by one, connecting vertices which are at large distances in the current graph. In comparison with the Erdos--Sachs proof, ours is slightly simpler while the value it achieves is slightly lower. Also, our algorithm works for all values of n and $k < \frac {n}{3}$, unlike most of the earlier constructions.
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TL;DR: This paper characterize the graphs with 1-hyperbolic metrics in terms of a convexity condition and forbidden isometric subgraphs.
Abstract: The shortest-path metric d of a graph G=(V,E) is called $\delta$-{\it hyperbolic} if for any four vertices $u,v,w,x\in X$ the two larger of the three sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most $\delta.$ In this paper, we characterize the graphs with 1-hyperbolic metrics in terms of a convexity condition and forbidden isometric subgraphs.
TL;DR: It is pointed out that the proofs imply that the MSSS problem is solvable in polynomial time for all digraphs that can be obtained from strong semicompleteDigraphs on at least two vertices by replacing each vertex with a digraph belonging to a family ofdigraphs whose path covering number can be decided in poynomial time.
Abstract: We consider the problem of finding a strongly connected spanning subdigraph with the minimum number of arcs in a strongly connected digraph. This problem is NP-hard for general digraphs since it generalizes the Hamiltonian cycle problem. We show that the problem is polynomially solvable for quasi-transitive digraphs. We describe the minimum number of arcs in such a spanning subdigraph of a quasi-transitive digraph in terms of the path covering number. Our proofs are based on a number of results (some of which are new and interesting in their own right) on the structure of cycles and paths in quasi-transitive digraphs and in extended semicomplete digraphs. In particular, we give a new characterization of the longest cycle in an extended semicomplete digraph. Finally, we point out that our proofs imply that the MSSS problem is solvable in polynomial time for all digraphs that can be obtained from strong semicomplete digraphs on at least two vertices by replacing each vertex with a digraph belonging to a family of digraphs whose path covering number can be decided in polynomial time.
TL;DR: A novel technique is given to compute a lower bound for the tree-width of a graph using maximum cardinality search, which is efficiently computable and guaranteed to do at least as well as finding the largest clique in the graph.
Abstract: The tree-width of a graph is of great importance in applied problems in graphical models. The complexity of inference problems on Markov random fields is exponential in the tree-width of the graph. However, computing tree-width is NP-hard in general. Easily computable upper bounds exist, but there are few lower bounds. We give a novel technique to compute a lower bound for the tree-width of a graph using maximum cardinality search. This bound is efficiently computable and is guaranteed to do at least as well as finding the largest clique in the graph.
TL;DR: In this article, it was shown that for arbitrary k, k-SUBCOLORABILITY can be decided in linear time on graphs with bounded treewidth and on graphs having bounded cliquewidth.
Abstract: In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erdos and, independently, Albertson et al., proved that every graph of maximum degree at most 3 has a 2-subcoloring. We point out that this fact is best possible with respect to degree constraints by showing that the problem of recognizing 2-subcolorable graphs with maximum degree 4 is NP-complete, even when restricted to triangle-free planar graphs. Moreover, in general, for fixed k, recognizing k-subcolorable graphs is NP-complete on graphs with maximum degree at most k2. In contrast, we show that, for arbitrary k, k-SUBCOLORABILITY can be decided in linear time on graphs with bounded treewidth and on graphs with bounded cliquewidth (including cographs as a specific case).
TL;DR: A modification of Adleman and Huang's method which runs conjecturally in expected time Lp, which is conjectured to compute a logarithm in a prime field whose cardinality p is of the form $r^e-s$, with r and s small in absolute value.
Abstract: Let p be a prime number and n a positive integer, and let q=pn. Adleman and Huang [Inform. and Comput., 151 (1999), pp. 5--16] have described a version of the function field sieve which is conjectured to compute a logarithm in the field of q elements in expected time Lq[1/3;(32/9)1/3+o(1)], where Lq[s;c]=exp(c(log q)s(log log q)1-s) and the o(1) is for $q\to\infty$ under the constraint that p6\leq n$. In this paper, we present a modification of their method which runs conjecturally in expected time Lq[1/3;(32/9)1/3+o(1)] so long as $q\to\infty$ with $p\leq n^{o(\sqrt{n})}$. The technique we use can also be applied to the special number field sieve and results in an algorithm which, in expected time Lp[1/3;(32/9)1/3+o(1)], is conjectured to compute a logarithm in a prime field whose cardinality p is of the form $r^e-s$, with r and s small in absolute value.
TL;DR: It is shown that the GRAPH 3-COLORABILITY problem remains NP-complete when restricted to linear dominoes with vertex degrees at most 4 and the existence of a finite list of forbidden induced subgraphs characterizing r is proved.
Abstract: A natural generalization of the notion of domino introduced and investigated in [T. Kloks, D. Kratsch, and H. Muller, Dominoes, Lecture Notes in Comput. Sci. 903, Springer-Verlag, Berlin, 1995, pp. 106--120] is considered. A graph is called an r-mino if each of its vertices belongs to at most r maximal cliques. The class of r-minoes is denoted ${\cal M}_r.$ Thus ${\cal M}_2$ is the class of dominoes. It is shown that ${\cal M}_r$ coincides with the class of line graphs of Helly hypergraphs with rank at most r. For an arbitrary r, the existence of a finite list of forbidden induced subgraphs characterizing ${\cal M}_r$ is proved. An explicit finite characterization is given for ${\cal M}_3$. An r-mino is called linear if each of its edges belongs to exactly one maximal clique. We prove that the GRAPH 3-COLORABILITY problem remains NP-complete when restricted to linear dominoes with vertex degrees at most 4.
IBM1
TL;DR: This work investigates the approximability of a no-wait permutation flow shop scheduling problem under the makespan criterion and presents a polynomial time approximation scheme (PTAS) for the problem on any fixed number of machines.
Abstract: We investigate the approximability of a no-wait permutation flow shop scheduling problem under the makespan criterion. We present a polynomial time approximation scheme (PTAS) for the problem on any fixed number of machines.
TL;DR: New methods are given to obtain facets from the subgraphs of the intersection graph associated with a set packing polyhedron that properly subsume several other methods in the literature.
Abstract: New results concerning the facial structure of set packing polyhedra are presented. In particular, new methods are given to obtain facets from the subgraphs of the intersection graph associated with a set packing polyhedron that properly subsume several other methods in the literature. A new class of facet defining graphs, termed fans, is also introduced, as well as a procedure to link any graph to a certain claw K1,k in order to obtain a new graph and an associated facet.
TL;DR: The main result is that this poset is a lattice that induces a partial order on this set that is isomorphic to the set of ordered partitions of a fixed multiset where the blocks are sets with a natural order relation.
Abstract: A Delannoy path is a minimal path with diagonal steps in ${\mathbb Z}^2$ between two arbitrary points. We extend this notion to the n dimensions space ${\mathbb Z}^n$ and identify such paths with words on a special kind of alphabet: an S-alphabet. We show that the set of all the words corresponding to Delannoy paths going from one point to another is exactly one class in the congruence generated by a Thue system that we exhibit. This Thue system induces a partial order on this set that is isomorphic to the set of ordered partitions of a fixed multiset where the blocks are sets with a natural order relation. Our main result is that this poset is a lattice.
TL;DR: It is proved that the class of linear threshold functions with polynomial-size weights can be divided into subclasses according to the degree of the polyn coefficients, and that there exists a minimal weight linear threshold function for any arbitrary number of inputs and any weight size.
Abstract: A linear threshold element computes a function that is a sign of a weighted sum of the input variables. The best known lower bounds on the size of threshold circuits are for depth-2 circuits with small (polynomial-size) weights. However, in general, the weights are arbitrary integers and can be of exponential size in the number of input variables. Namely, obtaining progress in lower bounds for threshold circuits seems to be related to understanding the role of large weights. In the present literature, a distinction is made between the two extreme cases of linear threshold functions with polynomial-size weights, as opposed to those with exponential-size weights. Our main contributions are in devising two novel methods for constructing threshold functions with minimal weights and filling up the gap between polynomial and exponential weight growth by further refining the separation. Namely, we prove that the class of linear threshold functions with polynomial-size weights can be divided into subclasses according to the degree of the polynomial. In fact, we prove a more general result---that there exists a minimal weight linear threshold function for any arbitrary number of inputs and any weight size.
TL;DR: Several binary linear codes are constructed which have best known optimal parameters, for instance, [20,11,5], [40,22,8], [48,21,12], and [72,31,16].
Abstract: We study certain projections of binary linear codes onto larger fields. These projections include the well-known projection of the extended Golay [24,12,8] code onto the hexacode over $\mbox{GF}(4)$ and the projection of the Reed--Muller code R(2,5) onto the unique self-dual [8,4,4] code over $\mbox{GF}(4)$. We give a characterization of these projections, and we construct several binary linear codes which have best known optimal parameters, for instance, [20,11,5], [40,22,8], [48,21,12], and [72,31,16]. We also relate the automorphism group of a quaternary code to that of the corresponding binary code.
TL;DR: The simplicial complex of t-colorable graphs on n vertices is proved to be homotopy equivalent to a wedge of spheres all of dimension $n(t-1)-\binom{t}{2}-1$ when t=2 and when $t\ge n-3$.
Abstract: We study the simplicial complex of t-colorable graphs on n vertices. We prove this complex is homotopy equivalent to a wedge of spheres all of dimension $n(t-1)-\binom{t}{2}-1$ when t=2 and when $t\ge n-3$. We show that such a homotopy equivalence does not hold for general t and n.
TL;DR: A large matrix A whose structure is simple, say, with unit entries on the first subdiagonal and superdiagonal is modified, and the eigenvalues and eigenvectors of this matrix are modified, finding the new eigenvalue from an elliptic integral.
Abstract: We start with a large matrix A whose structure is simple, say, with unit entries on the first subdiagonal and superdiagonal. Its eigenvalues and eigenvectors are known. We modify A in M widely spaced rows and columns. Then the "new eigenvectors" are nearly a sum of spikes xj = t|j-r| centered at the positions r of the modified rows. The new eigenvalues are given almost exactly by $\pm \sqrt{4+\mu^2}$, where $\mu$ is an eigenvalue of the M by M modification.
We extend this analysis to a larger class of structured matrices. For a banded Toeplitz matrix, our experiments show similar spikes centered around modified rows, and we have a conjecture on the structure of the new eigenvectors. For a single diagonal modification of the adjacency matrix of an infinite two-dimensional grid, we find the new eigenvalue from an elliptic integral (and we don't yet recognize the eigenvector).
TL;DR: A time randomized algorithm to approximate the length of the shortest vector in an n-dimensional lattice to within a factor of n^{3 + \epsilon}$.
Abstract: For every constant $\epsilon > 0$, we obtain a $2^{O(n(1/2 + 1/\epsilon))}$ time randomized algorithm to approximate the length of the shortest vector in an n-dimensional lattice to within a factor of $n^{3 + \epsilon}$.
TL;DR: It is shown that randomized algorithms can outperform deterministic algorithms, but only if the amount of work done is a nonconcave function of resource allocation.
Abstract: We consider a very general online scheduling problem with an objective to minimize the maximum level of resource allocated. We find a simple characterization of an optimal deterministic online algorithm. We develop further results for the two, more specific problems of single resource scheduling and hierarchical line balancing. We determine how to compute optimal online algorithms for both problems using linear programming and integer programming, respectively. We show that randomized algorithms can outperform deterministic algorithms, but only if the amount of work done is a nonconcave function of resource allocation.
TL;DR: The enumeration of labeled claw-free cubic graphs can be added to the handful of known counting problems for regular graphs with restrictions which have been proved P-recursive.
Abstract: Let Hn be the number of claw-free cubic graphs on 2n labeled nodes. Combinatorial reductions are used to derive a second order, linear homogeneous differential equation with polynomial coefficients whose power series solution is the exponential generating function for Hn. This leads to a recurrence relation for Hn which shows Hn to be P-recursive and which enables the sequence to be computed efficiently. Thus the enumeration of labeled claw-free cubic graphs can be added to the handful of known counting problems for regular graphs with restrictions which have been proved P-recursive.
TL;DR: A Fourier analysis for Radon transforms between multiplicity-free permutation representations is developed and applied to groups and Computation.
Abstract: We develop a Fourier analysis for Radon transforms between multiplicity-free permutation representations. Statistical applications of such Radon transforms were given by Diaconis and Rockmore in [Groups and Computation (New Brunswick, NJ, 1991), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 11, AMS, Providence, RI, 1993, pp. 87--104].
TL;DR: It is shown that convex geometries give rise to improved inclusion-exclusion identities and improved Bonferroni inequalities, which are applied in identifying a new class of hypergraphs for which the reliability covering problem can be solved in polynomial time.
Abstract: This paper establishes a connection between the theory of convex geometries, the principle of inclusion-exclusion, and the topological concept of an abstract tube. In particular, it is shown that convex geometries give rise to improved inclusion-exclusion identities and improved Bonferroni inequalities. In this way, several known results from the literature are rediscovered in a concise and unified way. The results are applied in identifying a new class of hypergraphs for which the reliability covering problem can be solved in polynomial time.
TL;DR: Bollobas and Hungar as discussed by the authors showed that the Farkas limit is the minimum of a certain mixed integer program, and showed how to compute it when each forbidden graph is a complete bipartite graph, in particular answering the question of Erdos, Faudree, Rousseau, and Schelp.
Abstract: We show that $\lim_{n\to\infty}\hat r(F_{1,n},\dots,F_{q,n},F_{q+1},\dots,F_{r})/n$ exists, where the bipartite graphs $F_{q+1},\dots,F_r$ do not depend on $n$ while, for $1\le i\le q$, $F_{i,n}$ is obtained from some bipartite graph $F_i$ with parts $V_1\cup V_2=V(F_i)$ by duplicating each vertex $v\in V_2$ $(c_v+o(1))n$ times for some real $c_v>0$.
In fact, the limit is the minimum of a certain mixed integer program. Using the Farkas lemma we show how to compute it when each forbidden graph is a complete bipartite graph, in particular answering the question of Erdos, Faudree, Rousseau, and Schelp [Period.\ Math.\ Hungar., 9 (1978), pp. 145--161], who asked for the asymptotics of $\hat r(K_{s,n},K_{s,n})$ for fixed $s$ and large $n$. Also, we prove (for all sufficiently large $n$) the conjecture of Faudree, Rousseau, and Sheehan in [Graph Theory and Combinatorics, B. Bollobas, ed., Cambridge University Press, Cambridge, UK, 1984, pp. 273--281] that $\hat r(K_{2,n},K_{2,n}) =18n-15$.