658 research outputs found
Graph coloring satisfying restraints
AbstractFor an integer k⩾2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k-colors. A graph G is amenably k-colorable if, for each nonconstant proper k-restraint r on G, there is a k-coloring c of G with c(v)≠r(v) for each vertex v of G. A graph G is amenable if it is amenably k-colorable and k is the chromatic number of G. For any k≠3, there are infinitely many amenable k-critical graphs. For k ⩾ 3, we use a construction of B. Toft and amenable graphs to associate a k-colorable graph to any k-colorable finite hypergraph. Some constructions for amenable graphs are given. We also consider a related property—being strongly critical—that is satisfied by many critical graphs, including complete graphs. A strongly critical graph is critical and amenable, but the converse is not always true. The Dirac join operation preserves both amenability and the strongly critical property. In addition, the Hajós construction applied to a single edge in each of two strongly k-critical graphs yields an amenable graph. However, for any k⩾5, there are amenable k-critical graphs for which the Hajós construction on two copies is not amenable
Ramsey-type graph coloring and diagonal non-computability
A function is diagonally non-computable (d.n.c.) if it diagonalizes against
the universal partial computable function. D.n.c. functions play a central role
in algorithmic randomness and reverse mathematics. Flood and Towsner asked for
which functions h, the principle stating the existence of an h-bounded d.n.c.
function (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper,
we prove that for every computable order h, there exists an~-model of
DNR_h which is not a not model of the Ramsey-type graph coloring principle for
two colors (RCOLOR2) and therefore not a model of RWKL. The proof combines
bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to
transform a computable non-reducibility into a separation over omega-models.Comment: 18 page
Optimal Joint Routing and Scheduling in Millimeter-Wave Cellular Networks
Millimeter-wave (mmWave) communication is a promising technology to cope with
the expected exponential increase in data traffic in 5G networks. mmWave
networks typically require a very dense deployment of mmWave base stations
(mmBS). To reduce cost and increase flexibility, wireless backhauling is needed
to connect the mmBSs. The characteristics of mmWave communication, and
specifically its high directional- ity, imply new requirements for efficient
routing and scheduling paradigms. We propose an efficient scheduling method,
so-called schedule-oriented optimization, based on matching theory that
optimizes QoS metrics jointly with routing. It is capable of solving any
scheduling problem that can be formulated as a linear program whose variables
are link times and QoS metrics. As an example of the schedule-oriented
optimization, we show the optimal solution of the maximum throughput fair
scheduling (MTFS). Practically, the optimal scheduling can be obtained even for
networks with over 200 mmBSs. To further increase the runtime performance, we
propose an efficient edge-coloring based approximation algorithm with provable
performance bound. It achieves over 80% of the optimal max-min throughput and
runs 5 to 100 times faster than the optimal algorithm in practice. Finally, we
extend the optimal and approximation algorithms for the cases of multi-RF-chain
mmBSs and integrated backhaul and access networks.Comment: To appear in Proceedings of INFOCOM '1
List rankings and on-line list rankings of graphs
A -ranking of a graph is a labeling of its vertices from
such that any nontrivial path whose endpoints have the same
label contains a larger label. The least for which has a -ranking is
the ranking number of , also known as tree-depth. The list ranking number of
is the least such that if each vertex of is assigned a set of
potential labels, then can be ranked by labeling each vertex with a label
from its assigned list. Rankings model a certain parallel processing problem in
manufacturing, while the list ranking version adds scheduling constraints. We
compute the list ranking number of paths, cycles, and trees with many more
leaves than internal vertices. Some of these results follow from stronger
theorems we prove about on-line versions of list ranking, where each vertex
starts with an empty list having some fixed capacity, and potential labels are
presented one by one, at which time they are added to the lists of certain
vertices; the decision of which of these vertices are actually to be ranked
with that label must be made immediately.Comment: 16 pages, 3 figure
On space efficiency of algorithms working on structural decompositions of graphs
Dynamic programming on path and tree decompositions of graphs is a technique
that is ubiquitous in the field of parameterized and exponential-time
algorithms. However, one of its drawbacks is that the space usage is
exponential in the decomposition's width. Following the work of Allender et al.
[Theory of Computing, '14], we investigate whether this space complexity
explosion is unavoidable. Using the idea of reparameterization of Cai and
Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely
related to a conjecture that the Longest Common Subsequence problem
parameterized by the number of input strings does not admit an algorithm that
simultaneously uses XP time and FPT space. Moreover, we complete the complexity
landscape sketched for pathwidth and treewidth by Allender et al. by
considering the parameter tree-depth. We prove that computations on tree-depth
decompositions correspond to a model of non-deterministic machines that work in
polynomial time and logarithmic space, with access to an auxiliary stack of
maximum height equal to the decomposition's depth. Together with the results of
Allender et al., this describes a hierarchy of complexity classes for
polynomial-time non-deterministic machines with different restrictions on the
access to working space, which mirrors the classic relations between treewidth,
pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new
version is augmented with a space-efficient algorithm for Dominating Set
using the Chinese remainder theore
On -stable locally checkable problems parameterized by mim-width
In this paper we continue the study of locally checkable problems under the
framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on
graphs of bounded mim-width. We study which restrictions on a locally checkable
problem are necessary in order to be able to solve it efficiently on graphs of
bounded mim-width. To this end, we introduce the concept of -stability of a
check function. The related locally checkable problems contain large classes of
problems, among which we can mention, for example, LCVP problems. We give an
algorithm showing that these problems are XP when parameterized by the
mim-width of a given binary decomposition tree of the input graph, that is,
that they can be solved in polynomial time given a binary decomposition tree of
bounded mim-width. We explore the relation between -stable locally checkable
problems and the recently introduced DN logic (Bergougnoux, Dreier and Jaffke,
2022), and show that both frameworks model the same family of problems. We
include a list of concrete examples of -stable locally checkable problems
whose complexity on graphs of bounded mim-width was open so far
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