A quasi-polynomial bound for the diameter of graphs of polyhedra

The diameter of the graph of a $d$-dimensional polyhedron with $n$ facets is at most $n^{\log d+2}$Comment: 2 page

An upper bound for the Ramsey numbers r(K3,G)

AbstractThe Ramsey number r(H,G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph KN in red and blue, it must contain either a ŕed H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K3,G)⩽2q+1 where G has q edges. In other words, any graph on 2q+1 vertices with independence number at most 2 contains every (isolate-free) graph on q edges. This establishes a 1980 conjecture of Harary. The result is best possible as a function of q

On the Number of Latent Subsets of Intersecting Collections

Not AvailableSupported in part by O.N.R. Contract N00014-67-A-0204-0016 and supported in part by the U.S. Army Research Office (Durham) under contract DAHCO4-70-C-0058

An algorithm for collapsing sign alternating sequences of real numbers

AbstractA table with two rows and n columns may be thought of as two vectors with n components. The distance between the two rows then corresponds to the norm of the difference between the rows. We examine the problem of how to collapse the adjacent columns of the table while keeping the norm of the difference as large as possible. First a stepwise algorithm is given which achieves this end with respect to the norm of the vector of differences. After proving the optimality of the stepwise solution we extend the result to the norm which arises from minimizing the number of persons misclassified. The same algorithm suffices

Point selections and weak e-nets for convex hulls

One of our results: let X be a finite set on the plane, 0 < g < 1, then there exists a set F (a weak g-net) of size at most 7/e 2 such that every convex set containing at least e\X\ elements of X intersects F. Note that the size of F is independent of the size of X. 1

Representations of families of triples over GF(2)

AbstractLet B be any family of 3-subsets of [n] = {1, …, n} such that every i in [n] belongs to at most three members of B. It is shown here that there exists a 3 × n (0, 1)-matrix M such that every set of columns of M indexed by a member of B is linearly independent over GF(2). The proof depends on finding a suitable vertex-coloring for the associated 3-uniform hypergraph. This matrix result, which is a special case of a conjecture of Griggs and Walker, implies the corresponding special case of a conjecture of Chung, Frankl, Graham, and Shearer and of Faudree, Schelp, and Sós concerning intersecting families of subsets

Systems of Linear Equations over F2\mathbb{F}_2 and Problems Parameterized Above Average

In the problem Max Lin, we are given a system $Az=b$ of $m$ linear equations with $n$ variables over $\mathbb{F}_2$ in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least $k$, where $k$ is the parameter. It is not hard to see that we may assume that no two equations in $Az=b$ have the same left-hand side and $n={\rm rank A}$. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: $m\le 2^{p(n)}$ for an arbitrary fixed function $p(n)=o(n)$. Max $r$-Lin AA is a special case of Max Lin AA, where each equation has at most $r$ variables. In Max Exact $r$-SAT AA we are given a multiset of $m$ clauses on $n$ variables such that each clause has $r$ variables and asked whether there is a truth assignment to the $n$ variables that satisfies at least $(1-2^{-r})m + k2^{-r}$ clauses. Using our maximum excess results, we prove that for each fixed $r\ge 2$, Max $r$-Lin AA and Max Exact $r$-SAT AA can be solved in time $2^{O(k \log k)}+m^{O(1)}.$ This improves $2^{O(k^2)}+m^{O(1)}$-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively