500 research outputs found
On the covering dimension of the set of solutions of some nonlinear equations
We prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose parametric equations are studied using cohomological mconclusions imply sharp, multidimensional continuation results. Applications are given to nonlinear partial differential equations
Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels
In this paper we study conditions which guarantee the existence of perfect
matchings and perfect fractional matchings in uniform hypergraphs. We reduce
this problem to an old conjecture by Erd\H{o}s on estimating the maximum number
of edges in a hypergraph when the (fractional) matching number is given, which
we are able to solve in some special cases using probabilistic techniques.
Based on these results, we obtain some general theorems on the minimum
-degree ensuring the existence of perfect (fractional) matchings. In
particular, we asymptotically determine the minimum vertex degree which
guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also
discuss an application to a problem of finding an optimal data allocation in a
distributed storage system
Approximating Minimum Cost Connectivity Orientation and Augmentation
We investigate problems addressing combined connectivity augmentation and
orientations settings. We give a polynomial-time 6-approximation algorithm for
finding a minimum cost subgraph of an undirected graph that admits an
orientation covering a nonnegative crossing -supermodular demand function,
as defined by Frank. An important example is -edge-connectivity, a
common generalization of global and rooted edge-connectivity.
Our algorithm is based on a non-standard application of the iterative
rounding method. We observe that the standard linear program with cut
constraints is not amenable and use an alternative linear program with
partition and co-partition constraints instead. The proof requires a new type
of uncrossing technique on partitions and co-partitions.
We also consider the problem setting when the cost of an edge can be
different for the two possible orientations. The problem becomes substantially
more difficult already for the simpler requirement of -edge-connectivity.
Khanna, Naor, and Shepherd showed that the integrality gap of the natural
linear program is at most when and conjectured that it is constant
for all fixed . We disprove this conjecture by showing an
integrality gap even when
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