In this paper, we investigate the impact of the broadcast effect arising in filterless optical networks on the computational complexity of the wavelength assignment problem. We model conflicts using an appropriate interference digraph, whose proper colourings correspond to feasible wavelength assignments. Minimizing the number of required wavelengths therefore amounts to determining the chromatic number of this interference digraph. Within this framework, we first present a polynomial-time 2-approximation algorithm for minimizing the number of wavelengths. We then show that the problem is fixed-parameter tractable when parameterized by the number k of available wavelengths. We also derive polynomial-time algorithms for computing the independence and clique numbers of this interference digraph.

The k-shortest simple paths problem asks to compute a set of top-k shortest simple paths from a source to a sink in a graph G = (V, E) with |V | = n vertices and |E| = m edges. The most wellknown algorithm for solving this problem is due to Yen (1971) with time complexity in (9(kn(m + nlog n)) and the fastest algorithm is due to Gotthilf and Lewenstein (2009) with time complexity in (9(kn(m + nlog logn)). For bounded treewidth graphs, Eppstein and Kurz (2017) lowered the computational complexity to (9(kn) by retrieving paths from the k smallest solutions of a monadic second-order formula, and to (9(n + k log(n)) to retrieve the k shortest simple distances only. In this paper, we provide an algorithm that answers k-shortest simple distances in (9(k + n) time on graphs with treewidth at most 2, and a constructive algorithm, simpler than that of Eppstein and Kurz, that solves the k-shortest simple paths problem in (9(kn) time on bounded treewidth graphs.

Cutwidth is a parameter used in many layout problems. Determining the cutwidth of a graph is an NP-complete problem, but it is possible to design efficient branch-and-bound algorithms if good lower bounds are available for cutting branches during exploration. Knowing how to quickly evaluate good bounds in each node of the search tree is therefore crucial.In this article, we give new lower bounds based on different graph density parameters such as the minimum, the average and the maximum average degree. Our main result is a new bound using the notion of traffic grooming on a path network, which appear to be in many cases better than bounds in the literature. Furthermore, the bound based on grooming can be computed quickly, in O(logn) time, and so is of interest to design faster branch-and-bound algorithms. Through extensive experiments, we show that this bound behaves very well compared to other bounds. Furthermore, we show how to obtain even better results when combining it with heuristics for finding dense subgraphs.

Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.

A beer graph is an edge-weighted graph G = ( V, E, omega) with beer vertices B subset of V. A beer path between two vertices s and t of a beer graph is a path that connects s and t and visits at least one vertex in B. The beer distance between two vertices is the weight of a shortest beer path, i.e. a beer path having minimum total weight. A graph indexing scheme is a two-phase method that constructs an index data structure by a one-time preprocessing of an input graph and then exploits it to compute (or accelerate the computation of) answers to queries on structures of the graph dataset. In the last decade, such indexing schemes have been designed to perform, effectively, many relevant types of queries, e.g. on reachability, and have gained significant popularity in essentially all data-intensive application domains where large number of queries have to be routinely answered (e.g. journey planners), since they have been shown, through many experimental studies, to offer extremely low query times at the price of limited preprocessing time and space overheads. In this paper, we showcase that an indexing scheme, to efficiently execute queries on beer distances or shortest beer paths for pairs of vertices of a beer graph, can be obtained by adapting the highway labeling, a recently introduced indexing method to accelerate the computation of classical shortest paths. We design a preprocessing algorithm to build a whl index, i.e. a weighted highway labeling of a beer graph, and show how it can be queried to compute beer distances and shortest beer paths. Through extensive experimentation on real networks, we empirically demonstrate its practical effectiveness and superiority, in terms of offered trade-off between preprocessing time, space overhead and query time, with respect to the state-of-the-art.

For any set system H=(V,R), R ⊆ 2^V, a subset S ⊆ V is called shattered if every S' ⊆ S results from the intersection of S with some set in . The VC-dimension of H is the size of a largest shattered set in V. In this paper, we focus on the problem of computing the VC-dimension of graphs. In particular, given a graph G=(V,E), the VC-dimension of G is defined as the VC-dimension of (V, 𝒩), where 𝒩 contains each subset of V that can be obtained as the closed neighborhood of some vertex v ∈ V in G. Our main contribution is an algorithm for computing the VC-dimension of any graph, whose effectiveness is shown through experiments on various types of practical graphs, including graphs with millions of vertices. A key aspect of its efficiency resides in the fact that practical graphs have small VC-dimension, up to 8 in our experiments. As a side-product, we present several new bounds relating the graph VC-dimension to other classical graph theoretical notions. We also establish the W[1]-hardness of the graph VC-dimension problem by extending a previous result for arbitrary set systems.

La largeur de coupe (cutwidth) est un param`etre utilis´e dans de nombreux probl`emes d’ordonnancement lin´eaire (layout). Calculer la largeur de coupe est un probl`eme NP-complet, mais il est possible de concevoir des algorithmes de branch-and-bound efficaces si on dispose de bonnes bornes inf´erieures pour couper des branches lors de l’exploration. Savoir ´evaluer rapidement de bonnes bornes dans chaque nœud de l’arbre de recherche est donc crucial. Dans cet article, nous donnons de nouvelles bornes bas´ees sur diff´erents param`etres de densit´e du graphe. Nous donnons aussi une borne inf´erieure bas´ee sur le groupage de requˆetes sur le chemin.

The k shortest simple path problem ( k SSP) asks to compute a set of top- k shortest simple paths from a source to a sink in a digraph. Yen (1971) proposed an algorithm with the best-known polynomial time complexity for this problem. Since then, the problem has been widely studied from an algorithm engineering perspective. The most noticeable proposals are the node-classification (NC) algorithm (Feng, 2014) and the sidetracks-based (SB) algorithm (Kurz, Mutzel, 2016). The latest offers the best running time at the price of a significant memory consumption. We first show how to speed up the SB algorithm using dynamic updates of shortest path trees resulting in a faster algorithm (SB*) with the same memory consumption. We then propose the parsimonious SB (PSB) algorithm that significantly reduces the memory consumption of SB at the cost of a small increase of the running time. Furthermore, we propose the postponed node-classification (PNC) algorithm that combines the best of the NC and the SB algorithms. It offers a significant speed up compared to the SB algorithm while using the same amount of memory as the NC algorithm. Our experimental results on complex networks show that all the considered algorithms have low memory consumption, and that the PSB algorithm is the fastest. On road networks, the relative performances of the algorithms depend on the number k of requested paths. Indeed, when the number k of requested paths is small (i.e., k ≤ 20 in our experiments), the SB* algorithm is the fastest among the considered algorithms, but it suffers from a large memory consumption and it offers very bad performances on some queries. When the number of requested paths is large (i.e., larger than 20 according to our experiments), the PNC algorithm is the fastest among the considered algorithms on road networks and it has a low memory footprint. The PNC algorithm is therefore a better choice on road networks.

Hyperbolicity is a graph parameter related to how much a graph resembles a tree with respect to distances. Its computation is challenging as the main approaches consist in scanning all quadruples of the graph or using fast matrix multiplication as building block, both are not practical for large graphs. In this paper, we propose and evaluate an approach that uses a hierarchy of distance-k dominating sets to reduce the search space. This technique, compared to the previous best practical algorithms, enables us to compute the hyperbolicity of graphs with unprecedented size (up to a million nodes) and speeds up the computation of previously attainable graphs by up to 3 orders of magnitude while reducing the memory consumption by up to more than a factor of 23.

Dynamic traffic leads to bandwidth fragmentation, which drastically reduces network performance, resulting in increased blocking rate and reduced bandwidth usage.When rerouting traffic flows at Layer 3 of an optical network, network operators are interested in minimizing the disturbances in order to satisfy their Service Level Agreements. Therefore, they turn to the Make-Before-Break (MBB) paradigm.In this paper, we revisit MBB rerouting with the objective of identifying the reroute sequence planning that minimizes the number of reroutes in order to minimize the resource usage.We propose a Dantzig-Wolfe decomposition mathematical model to solve this complex rerouting problem. We instigate how multiple or parallel rerouting reduces the overall minimum number of rerouting events (shortest makespan), and achieve the best resource usage. Numerical results bring interesting insights on that question and show a computational time reduction by about one order of magnitude over the state of the art.

Hyperbolicity is a graph parameter that indicates how much the shortest-path distance metric of a graph deviates from a tree metric. It is used in various fields such as networking, security, and bioinformatics for the classification of complex networks, the design of routing schemes, and the analysis of graph algorithms. Despite recent progress, computing the hyperbolicity of a graph remains challenging. Indeed, the best known algorithm has time complexity O(n 3.69 ) , which is prohibitive for large graphs, and the most efficient algorithms in practice have space complexity O(n 2 ) . Thus, time as well as space are bottlenecks for computing the hyperbolicity. In this article, we design a tool for enumerating all far-apart pairs of a graph by decreasing distances. A node pair (u, v) of a graph is far-apart if both v is a leaf of all shortest-path trees rooted at u and u is a leaf of all shortest-path trees rooted at v . This notion was previously used to drastically reduce the computation time for hyperbolicity in practice. However, it required the computation of the distance matrix to sort all pairs of nodes by decreasing distance, which requires an infeasible amount of memory already for medium-sized graphs. We present a new data structure that avoids this memory bottleneck in practice and for the first time enables computing the hyperbolicity of several large graphs that were far out of reach using previous algorithms. For some instances, we reduce the memory consumption by at least two orders of magnitude. Furthermore, we show that for many graphs, only a very small fraction of far-apart pairs has to be considered for the hyperbolicity computation, explaining this drastic reduction of memory. As iterating over far-apart pairs in decreasing order without storing them explicitly is a very general tool, we believe that our approach might also be relevant to other problems.

The similarity between two paths can be measured according to the proportion of arcs they share. We study the complexity of several variants of the problem of computing paths (whose measure of similarity does not exceed a certain threshold) between two given vertices of a weighted directed graph. For four of the most studied measures in the literature, we give a unified and simple proof of the fact that finding $k$ shortest dissimilar paths is NP-complete. We then consider the problem of finding an alternative to one or more given paths. We show that finding a path that is dissimilar to another given path can be done in polynomial time for one of the four considered measures while it is NP-complete for the three remaining measures. In addition, we show that if $k$ = 2 paths are given, finding a new path that is dissimilar to the given ones is NP-complete even on DAGs for the four considered measures. Moreover, for the four considered measures, we show that if a path $P$ is given, finding a shortest path among those that are dissimilar to P is NP-complete in DAGs.

The interactions between different brain regions can be modeled as a graph, called connectome, whose nodes correspond to parcels from a predefined brain atlas. The edges of the graph encode the strength of the axonal connectivity between regions of the atlas that can be estimated via diffusion magnetic resonance imaging (MRI) tractography. Herein, we aim to provide a novel perspective on the problem of choosing a suitable atlas for structural connectivity studies by assessing how robustly an atlas captures the network topology across different subjects in a homogeneous cohort. We measure this robustness by assessing the alignability of the connectomes, namely the possibility to retrieve graph matchings that provide highly similar graphs. We introduce two novel concepts. First, the graph Jaccard index (GJI), a graph similarity measure based on the well-established Jaccard index between sets; the GJI exhibits natural mathematical properties that are not satisfied by previous approaches. Second, we devise WL-align, a new technique for aligning connectomes obtained by adapting the Weisfeiler-Leman (WL) graph-isomorphism test. We validated the GJI and WL-align on data from the Human Connectome Project database, inferring a strategy for choosing a suitable parcellation for structural connectivity studies. Code and data are publicly available.

Journey planning in (schedule-based) public transit networks has attracted interest from researchers in the last decade.In particular, many algorithms aiming at efficiently answering queries of journey planning have been proposed.However, most of the proposed methods give the user a single or a limited number of journeys in practice, which is undesirable in a transportation context.In this paper, we consider the problem of finding k earliest arrival time journeys in public transit networks from a given origin to a given destination, i.e., an earliest arrival journey from the origin to the destination, a second earliest arrival journey, etc. until the k th earliest arrival journey.For this purpose, we propose an algorithm, denoted by Yen -Public Transit (Y-PT), which extends to public transit networks the algorithm proposed by Yen to find the top-k shortest simple paths in a graph.Moreover, we propose a more refined algorithm, called Postponed Yen -Public Transit (PY-PT), enabling a considerable speed up in practice.Our experiments on several public transit networks show that, in practice, PY-PT is faster than Y-PT by an order of magnitude.

The International Symposium on Experimental Algorithms (SEA) is an international forum for researchers in the area of the design, analysis, and experimental evaluation and engineering of algorithms, as well as in various aspects of computational optimization and its applications (telecommunications, transport, bioinformatics, cryptography, learning methods, etc.). The symposium aims at attracting papers from both the Computer Science and the Operations Research/Mathematical Programming communities. The main theme of the symposium is the role of experimentation and of algorithm engineering techniques in the design and evaluation of algorithms and data structures. Selected contributions to SEA present significant contributions supported by experimental evaluation, methodological issues in the design and interpretation of experiments, the use of heuristics and meta-heuritics, or application-driven case studies that deepen the understanding of the complexity of a problem. Each submission to SEA 2021 was reviewed by at least three Program Committee members or external reviewers. After a careful peer review and evaluation process, 23 papers were accepted for presentation and for inclusion in the proceedings, according to the reviewers’ recommendations. The scientific program of the symposium also includes presentations by three keynote speakers: Dominik Kempa (Johns Hopkins University, USA), Petra Mutzel (University of Bonn, Germany) and Blair D. Sullivan (University of Utah, USA). The 19th edition of the International Symposium on Experimental Algorithms (SEA 2021) was organized by the I3S laboratory (Universite Cote d’Azur, CNRS) with the help of Corinne Julien-Haddad. We thanks Universite Cote d’Azur, the research center Inria Sophia Antipolis - Mediterranee, and the city of Nice (Comite Doyen Lepine) for their financial support. We also thank the SEA steering committee for giving us the opportunity to host SEA 2021. We express our gratitude to the EasyChair platform. Thanks are also due to the editors of the ACM Journal of Experimental Algorithmics for their interest in hosting a special issue of the best papers presented at SEA 2021. Finally, we express our gratitude to the members of the Program Committee for their support, collaboration, and excellent work.

Filterless optical networks use passive splitters and combiners with coherent optics, providing wavelength selection in the digital domain, while forming a passive fiber-tree topology between nodes. In this study, we investigate the optimal design of filterless optical networks while minimizing the number of required wavelengths. We propose a Dantzig–Wolfe decomposition model in which each subproblem aims to generate a potential filterless optical subnetwork, with a directed tree topology. The master problem then selects the best combination of subnetworks. Numerical experiments demonstrate significant performance improvement over previous work, reducing previous computational results by a factor of 2 to 10, depending on the size of the data instances.

Optical multilayer optimization periodically reorganizes layer 0-1-2 network elements to handle both existing and dynamic traffic requirements in the most efficient manner. This delays the need for adding new resources in order to cope with the evolution of the traffic, thus saving CAPEX. The focus of this paper is on Layer 2, i.e., on capacity reoptimization at the optical transport network (OTN) layer when routes (e.g., LSPs in MPLS networks) are making unnecessarily long detours to evade congestion. Reconfiguration into optimized routes can be achieved by re-defining the routes, one at a time, so that they use the vacant resources generated by the disappearance of services using part of a path that transits the congested section. To maintain the Quality of Service, it is desirable to operate under a Make-Before-Break (MBB) paradigm, with the minimum number of reroutings. The challenge is to determine the best rerouting order while minimizing the bandwidth requirement. We propose an exact and scalable optimization model for computing a minimum bandwidth rerouting scheme subject to MBB in the OTN layer of an optical network. Numerical results show that we can successfully apply it on networks with up to 30 nodes, a very significant improvement with respect to the state of the art. We also provide some reoptimization analysis in terms of the bandwidth requirement vs. the number of reroutings.

Lorsque l'on demande a son GPS un chemin pour aller a La Rochelle, celui-ci propose generalement plusieurs chemins assez differents les uns des autres : l'un par l'autoroute, l'autre par la cote, un pas cher, etc. La notion de (dis)similarite entre deux chemins a ete definie, dans la litterature, de differentes manieres qui toutes sont liees a un ratio entre la longueur de leur intersection et une certaine fonction de leurs longueurs. Nous etudions la complexite de plusieurs variantes du probleme du calcul de chemins (dont la mesure de similarite n'excede pas un certain seuil) entre deux sommets d'un graphe oriente et pondere. Pour quatre des mesures les plus etudiees dans la litterature, nous donnons une preuve unifiee et simple du fait que trouver $k$ plus courts chemins dissimilaires est NP-complet. En pratique, ce que l'on cherche est une alternative a un ou des chemins que l'on connait a priori. Nos resultats principaux concernent ce type de probleme. Plus precisement, pour chacune des quatre mesures considerees, nous montrons que si $k\geq 2 $chemins sont donnes, en trouver un nouveau qui soit dissimilaire des premiers est NP-complet. Enfin, nous montrons que si un chemin $P$ est donne, trouver un plus court chemin parmi ceux qui sont dissimilaires de $P$ est NP-complet. Ce dernier resultat est a mettre en contraste avec le fait que, pour l'une des mesures, trouver un chemin dissimilaire a un chemin donne peut etre resolu tres simplement en temps polynomial.

Implementation in C++ of state-of-the-art algorithms for computing the k shortest simple paths from a source to a destination in a weighted directed graph.