15,474 research outputs found
Rendezvous of Distance-aware Mobile Agents in Unknown Graphs
We study the problem of rendezvous of two mobile agents starting at distinct
locations in an unknown graph. The agents have distinct labels and walk in
synchronous steps. However the graph is unlabelled and the agents have no means
of marking the nodes of the graph and cannot communicate with or see each other
until they meet at a node. When the graph is very large we want the time to
rendezvous to be independent of the graph size and to depend only on the
initial distance between the agents and some local parameters such as the
degree of the vertices, and the size of the agent's label. It is well known
that even for simple graphs of degree , the rendezvous time can be
exponential in in the worst case. In this paper, we introduce a new
version of the rendezvous problem where the agents are equipped with a device
that measures its distance to the other agent after every step. We show that
these \emph{distance-aware} agents are able to rendezvous in any unknown graph,
in time polynomial in all the local parameters such the degree of the nodes,
the initial distance and the size of the smaller of the two agent labels . Our algorithm has a time complexity of
and we show an almost matching lower bound of
on the time complexity of any
rendezvous algorithm in our scenario. Further, this lower bound extends
existing lower bounds for the general rendezvous problem without distance
awareness
On the Distinguishing Number of Cyclic Tournaments: Towards the Albertson-Collins Conjecture
A distinguishing -labeling of a digraph is a mapping from
the set of verticesof to the set of labels such that no
nontrivial automorphism of preserves all the labels.The distinguishing
number of is then the smallest for which admits a
distinguishing -labeling.From a result of Gluck (David Gluck, Trivial
set-stabilizers in finite permutation groups,{\em Can. J. Math.} 35(1) (1983),
59--67),it follows that for every cyclic tournament~ of (odd) order
.Let for every such tournament.Albertson and
Collins conjectured in 1999that the canonical 2-labeling given
by if and only if is distinguishing.We prove that
whenever one of the subtournaments of induced by vertices or
is rigid, satisfies Albertson-Collins Conjecture.Using
this property, we prove that several classes of cyclic tournaments satisfy
Albertson-Collins Conjecture.Moreover, we also prove that every Paley
tournament satisfies Albertson-Collins Conjecture
On tree-decompositions of one-ended graphs
A graph is one-ended if it contains a ray (a one way infinite path) and
whenever we remove a finite number of vertices from the graph then what remains
has only one component which contains rays. A vertex {\em dominates} a ray
in the end if there are infinitely many paths connecting to the ray such
that any two of these paths have only the vertex in common. We prove that
if a one-ended graph contains no ray which is dominated by a vertex and no
infinite family of pairwise disjoint rays, then it has a tree-decomposition
such that the decomposition tree is one-ended and the tree-decomposition is
invariant under the group of automorphisms.
This can be applied to prove a conjecture of Halin from 2000 that the
automorphism group of such a graph cannot be countably infinite and solves a
recent problem of Boutin and Imrich. Furthermore, it implies that every
transitive one-ended graph contains an infinite family of pairwise disjoint
rays
Optimal infinite scheduling for multi-priced timed automata
This paper is concerned with the derivation of infinite schedules for timed automata that are in some sense optimal. To cover a wide class of optimality criteria we start out by introducing an extension of the (priced) timed automata model that includes both costs and rewards as separate modelling features. A precise definition is then given of what constitutes optimal infinite behaviours for this class of models. We subsequently show that the derivation of optimal non-terminating schedules for such double-priced timed automata is computable. This is done by a reduction of the problem to the determination of optimal mean-cycles in finite graphs with weighted edges. This reduction is obtained by introducing the so-called corner-point abstraction, a powerful abstraction technique of which we show that it preserves optimal schedules
Automated Discharging Arguments for Density Problems in Grids
Discharging arguments demonstrate a connection between local structure and
global averages. This makes it an effective tool for proving lower bounds on
the density of special sets in infinite grids. However, the minimum density of
an identifying code in the hexagonal grid remains open, with an upper bound of
and a lower bound of . We present a new, experimental framework for producing discharging
arguments using an algorithm. This algorithm replaces the lengthy case analysis
of human-written discharging arguments with a linear program that produces the
best possible lower bound using the specified set of discharging rules. We use
this framework to present a lower bound of on
the density of an identifying code in the hexagonal grid, and also find several
sharp lower bounds for variations on identifying codes in the hexagonal,
square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables,
and 2 figure
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