30,298 research outputs found
Weighted k-Server Bounds via Combinatorial Dichotomies
The weighted -server problem is a natural generalization of the -server
problem where each server has a different weight. We consider the problem on
uniform metrics, which corresponds to a natural generalization of paging. Our
main result is a doubly exponential lower bound on the competitive ratio of any
deterministic online algorithm, that essentially matches the known upper bounds
for the problem and closes a large and long-standing gap.
The lower bound is based on relating the weighted -server problem to a
certain combinatorial problem and proving a Ramsey-theoretic lower bound for
it. This combinatorial connection also reveals several structural properties of
low cost feasible solutions to serve a sequence of requests. We use this to
show that the generalized Work Function Algorithm achieves an almost optimum
competitive ratio, and to obtain new refined upper bounds on the competitive
ratio for the case of different weight classes.Comment: accepted to FOCS'1
Large stars with few colors
A recent question in generalized Ramsey theory is that for fixed positive
integers , at least how many vertices can be covered by the vertices
of no more than monochromatic members of the family in every edge
coloring of with colors. This is related to an old problem of Chung
and Liu: for graph and integers what is the smallest positive
integer such that every coloring of the edges of with
colors contains a copy of with at most colors. We answer this question
when is a star and is either or generalizing the well-known
result of Burr and Roberts
Enumeration of small triangle free Ramsey Graphs
In 1930, a paper by Frank Plumpton Ramsey entitled On a Problem of Formal Logic appeared in the Proceedings of the London Mathematical Society. Although the impetus of this paper was one of mathematical logic, a far reaching combinatorial result was needed by Ramsey to achieve his objective. This combinatorial result became known as Ram \sey\u27s Theorem. One of the combinatorial structures which was developed during the study of Ramsey\u27s Theorem is that of a Ramsey graph. A Ramsey graph, denoted (k,l,n,e), is defined as an undirected graph that contains no cliques of size k, no independent sets of size I, with order n, and size e. Knowledge of Ramsey graphs is useful in the improvement of bounds and sometimes the calculation of exact values for various Ramsey number parameter situations. Straightforward enumeration of (k, I, n, e) Ramsey graphs for larger values of n is intractable with the current computing technology available. In order to produce such graphs, specialized algorithms need to be implemented. This thesis provides the theoretical background developed by Graver and Yackel [GRA68a], expanded upon by Grinstead and Roberts [GRl82a], and generalized by Radziszowski and Kreher [RAD88a, RAD88b] for the implementation of algorithms utilized for the enumeration of various Ram sey graphs. An object oriented graph manipulation package, including the above mentioned Ramsey graph enumeration algorithms, is implemented and documented. This package is utilized for the enumeration of all (3,3), (3,4), (3,5) and (3, 6) graphs. Some (3, 7) and (3, 8) also are calculated. These results duplicate and verify Ramsey graphs previously enumerated during other investigations. [RAD88a, RAD88b] In addition to these results, some newly enumerated (3,8) critical graphs, as well as some newly enumerated (3,9) graphs, including a minimum (3, 9, 26, 52) -graph are presented
Induced Ramsey-type results and binary predicates for point sets
Let and be positive integers and let be a finite point set in
general position in the plane. We say that is -Ramsey if there is a
finite point set such that for every -coloring of
there is a subset of such that and have the same order type
and is monochromatic in . Ne\v{s}et\v{r}il and Valtr proved
that for every , all point sets are -Ramsey. They also
proved that for every and , there are point sets that are
not -Ramsey.
As our main result, we introduce a new family of -Ramsey point sets,
extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result
to show that for every there is a point set such that no function
that maps ordered pairs of distinct points from to a set of size
can satisfy the following "local consistency" property: if attains
the same values on two ordered triples of points from , then these triples
have the same orientation. Intuitively, this implies that there cannot be such
a function that is defined locally and determines the orientation of point
triples.Comment: 22 pages, 3 figures, final version, minor correction
Independence Number and Disjoint Theta Graphs
The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θ-graph to be a pair of vertices u,v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k,α) of a graph with independence number α(G)≤α which contains no k disjoint θ-graphs. Since every θ-graph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k,α)+1. We also discuss the relationship between this problem and a generalized ramsey problem involving sets of graphs
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