8,853 research outputs found
Walking Through Waypoints
We initiate the study of a fundamental combinatorial problem: Given a
capacitated graph , find a shortest walk ("route") from a source to a destination that includes all vertices specified by a set
: the \emph{waypoints}. This waypoint routing problem
finds immediate applications in the context of modern networked distributed
systems. Our main contribution is an exact polynomial-time algorithm for graphs
of bounded treewidth. We also show that if the number of waypoints is
logarithmically bounded, exact polynomial-time algorithms exist even for
general graphs. Our two algorithms provide an almost complete characterization
of what can be solved exactly in polynomial-time: we show that more general
problems (e.g., on grid graphs of maximum degree 3, with slightly more
waypoints) are computationally intractable
Optimal Recombination in Genetic Algorithms
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results
Finding the connected components of the graph using perturbations of the adjacency matrix
The problem of finding the connected components of a graph is considered. The
algorithms addressed to solve the problem are used to solve such problems on
graphs as problems of finding points of articulation, bridges, maximin bridge,
etc. A natural approach to solving this problem is a breadth-first search, the
implementations of which are presented in software libraries designed to
maximize the use of the capabi\-lities of modern computer architectures. We
present an approach using perturbations of adjacency matrix of a graph. We
check wether the graph is connected or not by comparing the solutions of the
two systems of linear algebraic equations (SLAE): the first SLAE with a
perturbed adjacency matrix of the graph and the second SLAE with~unperturbed
matrix. This approach makes it possible to use effective numerical
implementations of SLAE solution methods to solve connectivity problems on
graphs. Iterations of iterative numerical methods for solving such SLAE can be
considered as carrying out a graph traversal. Generally speaking, the traversal
is not equivalent to the traversal that is carried out with breadth-first
search. An algorithm for finding the connected components of a graph using such
a traversal is presented. For any instance of the problem, this algorithm has
no greater computational complexity than breadth-first search, and for~most
individual problems it has less complexity.Comment: 22 pages, 4 figure
Percentile Queries in Multi-Dimensional Markov Decision Processes
Markov decision processes (MDPs) with multi-dimensional weights are useful to
analyze systems with multiple objectives that may be conflicting and require
the analysis of trade-offs. We study the complexity of percentile queries in
such MDPs and give algorithms to synthesize strategies that enforce such
constraints. Given a multi-dimensional weighted MDP and a quantitative payoff
function , thresholds (one per dimension), and probability thresholds
, we show how to compute a single strategy to enforce that for all
dimensions , the probability of outcomes satisfying is at least . We consider classical quantitative payoffs from
the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum,
discounted sum). Our work extends to the quantitative case the multi-objective
model checking problem studied by Etessami et al. in unweighted MDPs.Comment: Extended version of CAV 2015 pape
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