118 research outputs found
A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game
We prove a tight lower bound on the asymptotic performance ratio of
the bounded space online -hypercube bin packing problem, solving an open
question raised in 2005. In the classic -hypercube bin packing problem, we
are given a sequence of -dimensional hypercubes and we have an unlimited
number of bins, each of which is a -dimensional unit hypercube. The goal is
to pack (orthogonally) the given hypercubes into the minimum possible number of
bins, in such a way that no two hypercubes in the same bin overlap. The bounded
space online -hypercube bin packing problem is a variant of the
-hypercube bin packing problem, in which the hypercubes arrive online and
each one must be packed in an open bin without the knowledge of the next
hypercubes. Moreover, at each moment, only a constant number of open bins are
allowed (whenever a new bin is used, it is considered open, and it remains so
until it is considered closed, in which case, it is not allowed to accept new
hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448]
showed that is and , and conjectured that
it is . We show that is in fact . To
obtain this result, we elaborate on some ideas presented by those authors, and
go one step further showing how to obtain better (offline) packings of certain
special instances for which one knows how many bins any bounded space algorithm
has to use. Our main contribution establishes the existence of such packings,
for large enough , using probabilistic arguments. Such packings also lead to
lower bounds for the prices of anarchy of the selfish -hypercube bin packing
game. We present a lower bound of for the pure price of
anarchy of this game, and we also give a lower bound of for
its strong price of anarchy
The Size-Ramsey Number of 3-uniform Tight Paths
Given a hypergraph H, the size-Ramsey number Ër2(H) is the smallest integer m such that there exists a hypergraph G with m edges with the property that in any colouring of the edges of G with two colours there is a monochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices Pn(3) is linear in n, i.e., Ër2(Pn(3)) = O(n). This answers a question by Dudek, La Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417â434], who proved Ër2(Pn(3)) = O(n3/2 log3/2 n)
Partitioning by Monochromatic Trees
AbstractAnyr-edge-colouredn-vertex complete graphKncontains at mostrmonochromatic trees, all of different colours, whose vertex sets partition the vertex set ofKn, providednâ©Ÿ3r4r! (1â1/r)3(1âr)logr. This comes close to proving, for largen, a conjecture of Erdős, GyĂĄrfĂĄs, and Pyber, which states thatrâ1 trees suffice for alln
Near-optimum universal graphs for graphs with bounded degrees (Extended abstract)
Let H be a family of graphs. We say that G is H-universal if, for each H âH, the graph G contains a subgraph isomorphic to H. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an H(k, n)-universal graph Î(k, n) with O(n2â2/k(log n)1+8/k) edges. This is optimal up to a small polylogarithmic factor, as Ω(n2â2/k) is a lower bound for the number of edges in any such graph. En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Î(k, n), we prove, using a probabilistic argument, that Î(k, n) is H(k, n)-universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties. © Springer-Verlag Berlin Heidelberg 200
The size-Ramsey number of powers of paths
Given graphs and and a positive integer , say that
\emph{is -Ramsey for} , denoted
, if every -colouring of the edges of
contains a monochromatic copy of . The \emph{size-Ramsey number} \sr(H) of
a graph is defined to be
\sr(H)=\min\{|E(G)|\colon G\rightarrow (H)_2\}. Answering a
question of Conlon, we prove that, for every fixed~, we have
\sr(P_n^k)=O(n), where~ is the th power of the
-vertex path (i.e., the graph with vertex set and
all edges such that the distance between and in
is at most ). Our proof is probabilistic, but can also be made constructive.Most of the work for this paper was done during my PhD, which was half funded by EPSRC grant reference 1360036, and half by Merton College Oxford.
The third author was partially supported by FAPESP
(Proc.~2013/03447-6) and by CNPq (Proc.~459335/2014-6,
310974/2013-5). The fifth author was
supported by FAPESP (Proc.~2013/11431-2, Proc.~2013/03447-6 and
Proc.~2018/04876-1) and partially by CNPq (Proc.~459335/2014-6).
This research was supported in part by CAPES (Finance Code 001).
The collaboration of part of the authors was supported by a
CAPES/DAAD PROBRAL grant (Proc.~430/15)
Regularity inheritance in pseudorandom graphs
Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox, and Zhao. These inheritance lemmas also imply improved H-counting lemmas for subgraphs of bijumbled graphs, for some H
Quasirandom permutations are characterized by 4-point densities
For permutations Ï and Ï of lengths |Ï|â€|Ï| , let t(Ï,Ï) be the probability that the restriction of Ï to a random |Ï| -point set is (order) isomorphic to Ï . We show that every sequence {Ïj} of permutations such that |Ïj|ââ and t(Ï,Ïj)â1/4! for every 4-point permutation Ï is quasirandom (that is, t(Ï,Ïj)â1/|Ï|! for every Ï ). This answers a question posed by Graham
On two problems in graph Ramsey theory
We study two classical problems in graph Ramsey theory, that of determining
the Ramsey number of bounded-degree graphs and that of estimating the induced
Ramsey number for a graph with a given number of vertices.
The Ramsey number r(H) of a graph H is the least positive integer N such that
every two-coloring of the edges of the complete graph contains a
monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi
and Trotter states that there exists a constant c(\Delta) such that r(H) \leq
c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The
important open question is to determine the constant c(\Delta). The best
results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are
constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta
\log^2 \Delta}. We improve this upper bound, showing that there is a constant c
for which c(\Delta) \leq 2^{c \Delta \log \Delta}.
The induced Ramsey number r_{ind}(H) of a graph H is the least positive
integer N for which there exists a graph G on N vertices such that every
two-coloring of the edges of G contains an induced monochromatic copy of H.
Erd\H{o}s conjectured the existence of a constant c such that, for any graph H
on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this
conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an
earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in
the exponent.Comment: 18 page
Arithmetic progressions in sets of fractional dimension
Let E\subset\rr be a closed set of Hausdorff dimension . We prove
that if is sufficiently close to 1, and if supports a
probabilistic measure obeying appropriate dimensionality and Fourier decay
conditions, then contains non-trivial 3-term arithmetic progressions.Comment: 42 page
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