47 research outputs found
Simpler and Higher Lower Bounds for Shortcut Sets
We provide a variety of lower bounds for the well-known shortcut set problem:
how much can one decrease the diameter of a directed graph on vertices and
edges by adding or of shortcuts from the transitive closure
of the graph. Our results are based on a vast simplification of the recent
construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an
lower bound for the -sized shortcut set
problem. We highlight that our simplification completely removes the use of the
convex sets by B\'ar\'any and Larman [Math. Ann. 1998] used in all previous
lower bound constructions. Our simplification also removes the need for
randomness and further removes some log factors. This allows us to generalize
the construction to higher dimensions, which in turn can be used to show the
following results. For -sized shortcut sets, we show an
lower bound, improving on the previous best lower bound. For
all , we show that there exists a such that there
are -vertex -edge graphs where adding any shortcut set of size
keeps the diameter of at . This
improves the sparsity of the constructed graph compared to a known similar
result by Hesse [SODA 2003].
We also consider the sourcewise setting for shortcut sets: given a graph
, a set , how much can we decrease the sourcewise
diameter of , by adding a set of edges from the transitive closure of
? We show that for any integer , there exists a graph on
vertices and with ,
such that when adding or shortcuts, the sourcewise diameter is
.Comment: To appear in SODA 2024. Abstract shortened to fit arXiv requirement
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Energy Efficient Routing by Switching-Off Network Interfaces
International audienceSeveral studies exhibit that the traffic load of the routers only has a small influence on their energy consumption. Hence, the power consumption in networks is strongly related to the number of active network elements, such as interfaces, line cards, base chassis,... The goal thus is to find a routing that minimizes the (weighted) number of active network elements used when routing. In this paper, we consider a simplified architecture where a connection between two routers is represented as a link joining two network interfaces. When a connection is not used, both network interfaces can be turned off. Therefore, in order to reduce power consumption, the goal is to find the routing that minimizes the number of used links while satisfying all the demands. We first define formally the problem and we model it as an integer linear program. Then, we prove that this problem is not in APX, that is there is no polynomial-time constant-factor approximation algorithm. We propose a heuristic algorithm for this problem and we also prove some negative results about basic greedy and probabilistic algorithms. Thus we present a study on specific topologies, such as trees, grids and complete graphs, that provide bounds and results useful for real topologies. We then exhibit the gain in terms of number of network interfaces (leading to a global reduction of approximately 33 MWh for a medium-sized backbone network) for a set of existing network topologies: we see that for almost all topologies more than one third of the network interfaces can be spared for usual ranges of operation. Finally, we discuss the impact of energy efficient routing on the stretch factor and on fault tolerance.L'économie d'énergie dans les réseaux peut être accomplie en utilisant des techniques efficaces de routage ou de conception de réseaux. Dans ce papier, nous étudions une architecture simplifiée de réseaux dans laquelle lorsque deux routeurs sont reliés par un lien, les deux équipements extrémités de ce lien doivent être allumés. Chaque équipement ayant une consommation dépendant plutôt de son activation que de la quantité de traffic, notre objectif est de minimiser le nombre total d'équipements réseaux activés. Autrement dit, ce problème revient à effectuer un routage des demandes en minimisant le nombre d'arêtes dans la topologie. Nous proposons un programme linéaire pour résoudre ce problème et montrons des bornes simples sur des topologies particulières telles que la grille, l'arbre ou le graphe complet. Nous montrons des résultats d'inapproximabilité de ce problème, même si l'on considère des instances particulières. Nous proposons ensuite une heuristique dont nous évaluons les performances à l'aide de simulations sur des topologies réelles et sur la grille. Nous étudions ensuite l'impact de ces solutions efficaces en énergie sur la tolérance aux pannes et sur la longueur moyenne des routes. Finalement, nous proposons des structures de routage qui garantissent deux chemins disjoints par demande, ainsi qu'une limite sur la longueur des chemins
A PTAS for Euclidean TSP with Hyperplane Neighborhoods
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given
a collection of geometric regions in some space. The goal is to output a tour
of minimum length that visits at least one point in each region. Even in the
Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying
more tractable special cases of the problem. In this paper, we focus on the
fundamental special case of regions that are hyperplanes in the -dimensional
Euclidean space. This case contrasts the much-better understood case of
so-called fat regions.
While for an exact algorithm with running time is known,
settling the exact approximability of the problem for has been repeatedly
posed as an open question. To date, only an approximation algorithm with
guarantee exponential in is known, and NP-hardness remains open.
For arbitrary fixed , we develop a Polynomial Time Approximation Scheme
(PTAS) that works for both the tour and path version of the problem. Our
algorithm is based on approximating the convex hull of the optimal tour by a
convex polytope of bounded complexity. Such polytopes are represented as
solutions of a sophisticated LP formulation, which we combine with the
enumeration of crucial properties of the tour. As the approximation guarantee
approaches , our scheme adjusts the complexity of the considered polytopes
accordingly.
In the analysis of our approximation scheme, we show that our search space
includes a sufficiently good approximation of the optimum. To do so, we develop
a novel and general sparsification technique to transform an arbitrary convex
polytope into one with a constant number of vertices and, in turn, into one of
bounded complexity in the above sense. Hereby, we maintain important properties
of the polytope
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum