12,519 research outputs found
Shortest paths in nearly conservative digraphs
We introduce the following notion: a digraph D = (V, A) with arc weights c: A â R is called nearly conservative if every negative cycle consists of two arcs. Computing shortest paths in nearly conservative digraphs is NP-hard, and even deciding whether a digraph is nearly conservative is coNP-complete. We show that the âAll Pairs Shortest Pathâ problem is fixed parameter tractable with various parameters for nearly conservative digraphs. The results also apply for the special case of conservative mixed graphs
Shortest Odd Paths in Undirected Graphs with Conservative Weight Functions
We consider the Shortest Odd Path problem, where given an undirected graph
, a weight function on its edges, and two vertices and in , the
aim is to find an -path with odd length and, among all such paths, of
minimum weight. For the case when the weight function is conservative, i.e.,
when every cycle has non-negative total weight, the complexity of the Shortest
Odd Path problem had been open for 20 years, and was recently shown to be
NP-hard. We give a polynomial-time algorithm for the special case when the
weight function is conservative and the set of negative-weight edges
forms a single tree. Our algorithm exploits the strong connection between
Shortest Odd Path and the problem of finding two internally vertex-disjoint
paths between two terminals in an undirected edge-weighted graph. It also
relies on solving an intermediary problem variant called Shortest
Parity-Constrained Odd Path where for certain edges we have parity constraints
on their position along the path. Also, we exhibit two FPT algorithms for
solving Shortest Odd Path in graphs with conservative weight functions. The
first FPT algorithm is parameterized by , the number of negative edges,
or more generally, by the maximum size of a matching in the subgraph of
spanned by . Our second FPT algorithm is parameterized by the treewidth of
On Maltsev Digraphs
This is an Open Access article, first published by E-CJ on 25 February 2015.We study digraphs preserved by a Maltsev operation: Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing in this way that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. We then generalize results from Kazda (2011) to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a O(|VG|4)-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation, and relate them with series parallel digraphs.Peer reviewedFinal Published versio
Weighted distances in scale-free preferential attachment models
We study three preferential attachment models where the parameters are such
that the asymptotic degree distribution has infinite variance. Every edge is
equipped with a non-negative i.i.d. weight. We study the weighted distance
between two vertices chosen uniformly at random, the typical weighted distance,
and the number of edges on this path, the typical hopcount. We prove that there
are precisely two universality classes of weight distributions, called the
explosive and conservative class. In the explosive class, we show that the
typical weighted distance converges in distribution to the sum of two i.i.d.
finite random variables. In the conservative class, we prove that the typical
weighted distance tends to infinity, and we give an explicit expression for the
main growth term, as well as for the hopcount. Under a mild assumption on the
weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear
in Random Structures and Algorithm
Adaptive Probabilistic Flooding for Multipath Routing
In this work, we develop a distributed source routing algorithm for topology
discovery suitable for ISP transport networks, that is however inspired by
opportunistic algorithms used in ad hoc wireless networks. We propose a
plug-and-play control plane, able to find multiple paths toward the same
destination, and introduce a novel algorithm, called adaptive probabilistic
flooding, to achieve this goal. By keeping a small amount of state in routers
taking part in the discovery process, our technique significantly limits the
amount of control messages exchanged with flooding -- and, at the same time, it
only minimally affects the quality of the discovered multiple path with respect
to the optimal solution. Simple analytical bounds, confirmed by results
gathered with extensive simulation on four realistic topologies, show our
approach to be of high practical interest.Comment: 6 pages, 6 figure
Transit Node Routing Reconsidered
Transit Node Routing (TNR) is a fast and exact distance oracle for road
networks. We show several new results for TNR. First, we give a surprisingly
simple implementation fully based on Contraction Hierarchies that speeds up
preprocessing by an order of magnitude approaching the time for just finding a
CH (which alone has two orders of magnitude larger query time). We also develop
a very effective purely graph theoretical locality filter without any
compromise in query times. Finally, we show that a specialization to the online
many-to-one (or one-to-many) shortest path further speeds up query time by an
order of magnitude. This variant even has better query time than the fastest
known previous methods which need much more space.Comment: 19 pages, submitted to SEA'201
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