57 research outputs found
On a problem of Yuzvinsky on separating the n-cube
AbstractThe following problem of Yuzvinsky is solved here: how many vertices of the n-cube must be removed from it in order that no connected component of the rest contains an antipodal pair of vertices? Some further results and problems are described as well
The number of meets between two subsets of a lattice
AbstractLet L be a lattice of divisors of an integer (isomorphically, a direct product of chains). We prove |A| |B| ⩽ |L| |A ∧ B| for any A, B ⊃ L, where |·| denotes cardinality and A ∧ B = {a ∧ b: a ϵ A, b ϵ B}. |A ∧ B| attains its minimum for fixed |A|, |B| when A and B are ideals. |·| can be replaced by certain other weight functions. When the n chains are of equal size k, the elements may be viewed as n-digit k-ary numbers. Then for fixed |A|, |B|, |A ∧ B| is minimized when A and B are the |A| and |B| smallest n-digit k-ary numbers written backwards and forwards, respectively. |A ∧ B| for these sets is determined and bounded. Related results are given, and conjectures are made
Systems of Linear Equations over and Problems Parameterized Above Average
In the problem Max Lin, we are given a system of linear equations
with variables over 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 , where is the parameter.
It is not hard to see that we may assume that no two equations in have
the same left-hand side and . Using our maximum excess results,
we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable
for a wide special case: for an arbitrary fixed function
.
Max -Lin AA is a special case of Max Lin AA, where each equation has at
most variables. In Max Exact -SAT AA we are given a multiset of
clauses on variables such that each clause has variables and asked
whether there is a truth assignment to the variables that satisfies at
least clauses. Using our maximum excess results, we
prove that for each fixed , Max -Lin AA and Max Exact -SAT AA can
be solved in time This improves
-time algorithms for the two problems obtained by Gutin et
al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively
On potentially -graphic sequences
summary:For given a graph , a graphic sequence is said to be potentially -graphic if there is a realization of containing as a subgraph. In this paper, we characterize the potentially -positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence to be potentially -graphic, where is a complete graph on vertices and is a graph obtained from by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence to be potentially -graphic
An inequality for the weights of two families of sets, their unions and intersections
Ahlswede R, Daykin DE. An inequality for the weights of two families of sets, their unions and intersections. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 1978;43(3):183-185
- …