45 research outputs found
Randomized Speedup of the Bellman-Ford Algorithm
We describe a variant of the Bellman-Ford algorithm for single-source
shortest paths in graphs with negative edges but no negative cycles that
randomly permutes the vertices and uses this randomized order to process the
vertices within each pass of the algorithm. The modification reduces the
worst-case expected number of relaxation steps of the algorithm, compared to
the previously-best variant by Yen (1970), by a factor of 2/3 with high
probability. We also use our high probability bound to add negative cycle
detection to the randomized algorithm.Comment: 12 Pages, 6 Figures, ANALCO 201
Using Strategy Improvement to Stay Alive
We design a novel algorithm for solving Mean-Payoff Games (MPGs). Besides
solving an MPG in the usual sense, our algorithm computes more information
about the game, information that is important with respect to applications. The
weights of the edges of an MPG can be thought of as a gained/consumed energy --
depending on the sign. For each vertex, our algorithm computes the minimum
amount of initial energy that is sufficient for player Max to ensure that in a
play starting from the vertex, the energy level never goes below zero. Our
algorithm is not the first algorithm that computes the minimum sufficient
initial energies, but according to our experimental study it is the fastest
algorithm that computes them. The reason is that it utilizes the strategy
improvement technique which is very efficient in practice
Pareto Optimal Matchings in Many-to-Many Markets with Ties
We consider Pareto-optimal matchings (POMs) in a many-to-many market of
applicants and courses where applicants have preferences, which may include
ties, over individual courses and lexicographic preferences over sets of
courses. Since this is the most general setting examined so far in the
literature, our work unifies and generalizes several known results.
Specifically, we characterize POMs and introduce the \emph{Generalized Serial
Dictatorship Mechanism with Ties (GSDT)} that effectively handles ties via
properties of network flows. We show that GSDT can generate all POMs using
different priority orderings over the applicants, but it satisfies truthfulness
only for certain such orderings. This shortcoming is not specific to our
mechanism; we show that any mechanism generating all POMs in our setting is
prone to strategic manipulation. This is in contrast to the one-to-one case
(with or without ties), for which truthful mechanisms generating all POMs do
exist
Pareto optimal matchings in many-to-many markets with ties
We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants
and courses where applicants have preferences, which may include ties, over
individual courses and lexicographic preferences over sets of courses. Since this is the
most general setting examined so far in the literature, our work unifies and generalizes
several known results. Specifically, we characterize POMs and introduce the Generalized
Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties
via properties of network flows. We show that GSDT can generate all POMs using
different priority orderings over the applicants, but it satisfies truthfulness only for
certain such orderings. This shortcoming is not specific to our mechanism; we show
that any mechanism generating all POMs in our setting is prone to strategic manipulation.
This is in contrast to the one-to-one case (with or without ties), for which
truthful mechanisms generating all POMs do exist
Engineering Negative Cycle Canceling for Wind Farm Cabling
In a wind farm turbines convert wind energy into electrical energy. The generation of each turbine is transmitted, possibly via other turbines, to a substation that is connected to the power grid. On every possible interconnection there can be at most one of various different cable types. Each cable type comes with a cost per unit length and with a capacity. Designing a cost-minimal cable layout for a wind farm to feed all turbine production into the power grid is called the Wind Farm Cabling Problem (WCP). We consider a formulation of WCP as a flow problem on a graph where the cost of a flow on an edge is modeled by a step function originating from the cable types. Recently, we presented a proof-of-concept for a negative cycle canceling-based algorithm for WCP [Sascha Gritzbach et al., 2018]. We extend key steps of that heuristic and build a theoretical foundation that explains how this heuristic tackles the problems arising from the special structure of WCP. A thorough experimental evaluation identifies the best setup of the algorithm and compares it to existing methods from the literature such as Mixed-integer Linear Programming (MILP) and Simulated Annealing (SA). The heuristic runs in a range of half a millisecond to under two minutes on instances with up to 500 turbines. It provides solutions of similar quality compared to both competitors with running times of one hour and one day. When comparing the solution quality after a running time of two seconds, our algorithm outperforms the MILP- and SA-approaches, which allows it to be applied in interactive wind farm planning