8,700 research outputs found
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs
How Long It Takes for an Ordinary Node with an Ordinary ID to Output?
In the context of distributed synchronous computing, processors perform in
rounds, and the time-complexity of a distributed algorithm is classically
defined as the number of rounds before all computing nodes have output. Hence,
this complexity measure captures the running time of the slowest node(s). In
this paper, we are interested in the running time of the ordinary nodes, to be
compared with the running time of the slowest nodes. The node-averaged
time-complexity of a distributed algorithm on a given instance is defined as
the average, taken over every node of the instance, of the number of rounds
before that node output. We compare the node-averaged time-complexity with the
classical one in the standard LOCAL model for distributed network computing. We
show that there can be an exponential gap between the node-averaged
time-complexity and the classical time-complexity, as witnessed by, e.g.,
leader election. Our first main result is a positive one, stating that, in
fact, the two time-complexities behave the same for a large class of problems
on very sparse graphs. In particular, we show that, for LCL problems on cycles,
the node-averaged time complexity is of the same order of magnitude as the
slowest node time-complexity.
In addition, in the LOCAL model, the time-complexity is computed as a worst
case over all possible identity assignments to the nodes of the network. In
this paper, we also investigate the ID-averaged time-complexity, when the
number of rounds is averaged over all possible identity assignments. Our second
main result is that the ID-averaged time-complexity is essentially the same as
the expected time-complexity of randomized algorithms (where the expectation is
taken over all possible random bits used by the nodes, and the number of rounds
is measured for the worst-case identity assignment).
Finally, we study the node-averaged ID-averaged time-complexity.Comment: (Submitted) Journal versio
Decomposition, Reformulation, and Diving in University Course Timetabling
In many real-life optimisation problems, there are multiple interacting
components in a solution. For example, different components might specify
assignments to different kinds of resource. Often, each component is associated
with different sets of soft constraints, and so with different measures of soft
constraint violation. The goal is then to minimise a linear combination of such
measures. This paper studies an approach to such problems, which can be thought
of as multiphase exploitation of multiple objective-/value-restricted
submodels. In this approach, only one computationally difficult component of a
problem and the associated subset of objectives is considered at first. This
produces partial solutions, which define interesting neighbourhoods in the
search space of the complete problem. Often, it is possible to pick the initial
component so that variable aggregation can be performed at the first stage, and
the neighbourhoods to be explored next are guaranteed to contain feasible
solutions. Using integer programming, it is then easy to implement heuristics
producing solutions with bounds on their quality.
Our study is performed on a university course timetabling problem used in the
2007 International Timetabling Competition, also known as the Udine Course
Timetabling Problem. In the proposed heuristic, an objective-restricted
neighbourhood generator produces assignments of periods to events, with
decreasing numbers of violations of two period-related soft constraints. Those
are relaxed into assignments of events to days, which define neighbourhoods
that are easier to search with respect to all four soft constraints. Integer
programming formulations for all subproblems are given and evaluated using ILOG
CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table
Node Labels in Local Decision
The role of unique node identifiers in network computing is well understood
as far as symmetry breaking is concerned. However, the unique identifiers also
leak information about the computing environment - in particular, they provide
some nodes with information related to the size of the network. It was recently
proved that in the context of local decision, there are some decision problems
such that (1) they cannot be solved without unique identifiers, and (2) unique
node identifiers leak a sufficient amount of information such that the problem
becomes solvable (PODC 2013).
In this work we give study what is the minimal amount of information that we
need to leak from the environment to the nodes in order to solve local decision
problems. Our key results are related to scalar oracles that, for any given
, provide a multiset of labels; then the adversary assigns the
labels to the nodes in the network. This is a direct generalisation of the
usual assumption of unique node identifiers. We give a complete
characterisation of the weakest oracle that leaks at least as much information
as the unique identifiers.
Our main result is the following dichotomy: we classify scalar oracles as
large and small, depending on their asymptotic behaviour, and show that (1) any
large oracle is at least as powerful as the unique identifiers in the context
of local decision problems, while (2) for any small oracle there are local
decision problems that still benefit from unique identifiers.Comment: Conference version to appear in the proceedings of SIROCCO 201
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