International audience

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 well-known algorithm for solving this problem is due to Yen (1971) with time complexity in O ( k n ( m + n log ⁡ n ) ) and the fastest algorithm is due to Gotthilf and Lewenstein (2009) with time complexity in O ( k n ( m + n log ⁡ log ⁡ n ) ) . For bounded treewidth graphs, Eppstein and Kurz (2017) lowered the computational complexity to O ( k n ) by retrieving paths from the k smallest solutions of a monadic second-order formula, and to O ( 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 O ( 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 O ( k n ) time on bounded treewidth graphs.

A beer graph is an edge-weighted graph G = (V,E,ω) with beer vertices B ⊆ 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 ℛ. 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.

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).

International audience

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.

The k shortest simple path problem (kSSP) asks to compute a set of top-k shortest simple paths from a vertex s to a vertex t in a digraph. Yen (1971) proposed the first algorithm with the best known theoretical complexity of O(kn(m+n log n)) for a digraph with n vertices and m arcs. Since then, the problem has been widely studied from an algorithm engineering perspective, and impressive improvements have been achieved. In particular, Kurz and Mutzel (2016) proposed a sidetracks-based (SB) algorithm which is currently the fastest solution. In this work, we propose two improvements of this algorithm. We first show how to speed up the SB algorithm using dynamic updates of shortest path trees. We did experiments on some road networks of the 9th DIMAC'S challenge with up to about half a million nodes and one million arcs. Our computational results show an average speed up by a factor of 1.5 to 2 with a similar working memory consumption as SB. We then propose a second algorithm enabling to significantly reduce the working memory at the cost of an increase of the running time (up to two times slower). Our experiments on the same data set show, on average, a reduction by a factor of 1.5 to 2 of the working memory.

The use of drones has become more widespread in recent years. Many use cases have developed involving these autonomous vehicles, ranging from simple delivery of packages to complex emergency situations following catastrophic events. The miniaturization and very low cost of these machines make it possible today to create large meshes to ensure network coverage in disaster areas, for instance. However, the problems of scaling up and self-organization are necessary to solve problems in these use cases. This position paper first presents different new requirements for the deployment of unmanned aerial vehicles (UAV) networks, involving the use of many drones. Then, it introduces solutions from distributed algorithms and real-time data processing to ensure quasi-optimal solutions to the raised problems.

Recently, hardness results for problems in P were achieved using reasonable complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph-theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above-mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width , an important graph parameter, remained to be done. In this article, we study several graph-theoretic problems for which hardness results exist such as cycle problems , distance problems , and maximum matching . We give hardness results and P-FPT algorithms, using clique-width and some of its upper bounds as parameters. We believe that our most important result is an algorithm in O ( k 4 ⋅ n + m )-time for computing a maximum matching, where k is either the modular-width of the graph or the P 4 -sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

Optical multilayer optimization continuously reorganizes layer 0-1-2 network elements to handle both existing and dynamic traffic requirements in the most efficient manner. This delays the need to add new resources for new requests, saving CAPEX and leads to optical network defragmentation. The focus of this paper is on Layer 2, i.e., on capacity defragmentation at the 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. For the Quality of Service, it is desirable to operate under Make Before Break (MBB), with the minimum number of rerouting. The challenge is to identify the rerouting order, one connection at a time, 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 the state of the art. We also provide some defragmentation analysis in terms of the bandwidth requirement vs. the number of reroutings.

Recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width, an important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems, distance problems and maximum matching. We give hardness results and P-FPT algorithms, using clique-width and some of its upper-bounds as parameters. We believe that our most important result is an O(k4 · n + m)-time algorithm for computing a maximum matching where k is either the modular-width or the P4-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

We study the complexity of decomposing a graph by means of clique separators. This common algorithmic tool, first introduced by Tarjan, allows one to cut a graph into smaller pieces, and so it can be applied to preprocess the graph in the computation of optimization problems. However, the best-known algorithms for computing a decomposition have respective ${\cal O}(nm)$-time and ${\cal O}(n^{(3+\alpha)/2}) = o(n^{2.69})$-time complexity with $\alpha < 2.3729$ being the exponent for matrix multiplication. Such running times are prohibitive for large graphs. Here we prove that for every graph $G$, a decomposition can be computed in ${\cal O}(T(G) + \min\{n^{\alpha},\omega^2 n\})$-time with $T(G)$ and $\omega$ being, respectively, the time needed to compute a minimal triangulation of $G$ and the clique-number of $G$. In particular, it implies that every graph can be decomposed by clique separators in ${\cal O}(n^{\alpha}\log n)$-time. Based on prior work from Kratsch et al., we prove in addition that decomposing a graph by clique-separators is as least as hard as triangle detection. Therefore, the existence of any $o(n^{\alpha})$-time algorithm for this problem would be a significant breakthrough in the algorithmic field. Finally, our main result implies that planar graphs, bounded-treewidth graphs, and bounded-degree graphs can be decomposed by clique separators in linear or quasi-linear time.

Tree-likeness parameters have proven their utility in the design of efficient algorithms on graphs. In this paper, we relate the structural tree-likeness of graphs with their metric tree-likeness. To this end, we establish new upper bounds on the diameter of minimal separators in graphs. We prove that in any graph $G$, the diameter of any minimal separator $S$ in $G$ is at most $\lfloor \ell(G) / 2\rfloor \cdot (|S|-1)$, with $\ell(G)$ the length of a longest isometric cycle in $G$. Our result relies on algebraic methods and on the cycle basis of graphs. We improve our bound for the graphs admitting a distance preserving elimination ordering, for which we prove that any minimal separator $S$ has diameter at most $2 \cdot (|S|-1)$. We use our results to prove that the treelength $tl(G)$ of any graph $G$ is at most $\lfloor \ell(G) / {2}\rfloor$ times its treewidth $tw(G)$. In addition, we prove that, for any graph $G$ that excludes an apex graph $H$ as a minor, $tw(G) \leq c_H \cdot tl(G)$ for some constant $c_H$ only depending on $H$. We refine this constant when $G$ has bounded genus. Altogether, we obtain a simple $\mathcal{O} (\ell(G))$-approximation algorithm for computing the treewidth of $n$-node apex-minor-free graphs in $\mathcal{O}(n^2)$-time.

We aim at dimensioning fixed broadband microwave wireless networks under unreliable channel conditions. As the transport capacity of microwave links is prone to variations due to, e.g., weather conditions, such a dimensioning requires special attention. It can be formulated as the determination of the minimum cost bandwidth assignment of the links in the network for which traffic requirements can be met with high probability, while taking into account that transport link capacities vary depending on channel conditions. The proposed optimization model represents a major step forward since we consider dynamic routing. Experimental results show that the resulting solutions can save up to 45% of the bandwidth cost compared to the case where a bandwidth over-provisioning policy is uniformly applied to all links in the network planning. Comparisons with previous work also show that we can solve much larger instances in significantly shorter computing times, with a comparable level of reliability.

In this paper, we extend our former investigation on conceiving reliable fixed point-to-point wireless networks under outage probability constraints. We consider the problem of determining the minimum cost bandwidth assignment of a network, while guaranteeing a reliability level of the solution. If the optimal bandwidth assignment and routing of traffic demands are accomplished, the reliability criterion requires that network flows remain feasible with high probability, regarding that the performance of microwave links is prone to variations due to external factors, e.g., weather. We introduce a chance-constrained programming approach to tackle this problem and we present reformulations to standard integer linear programming models, including a budget-constrained formulation. To improve the solving performance, we propose new valid inequalities and a primal heuristic. Computational results present a performance analysis of the valid inequalities and the heuristic. Further, the outperformance of the novel model compared to more traditional approaches is documented.

The shortest-path metric ${\textup{d}}$ of a connected graph $G$ is ${1}/{2}$-hyperbolic if and only if it satisfies ${\textup{d}}(u,v) + {\textup{d}}(x,y) \leq \max \{ {\textup{d}}(u,x) + {\textup{d}}(v,y), {\textup{d}}(u,y) + {\textup{d}}(v,x) \} + 1$, for every $4$-tuple $u$, $x$, $v$, $y$ of $G$. We show that the problem of deciding whether an unweighted graph is ${1}/{2}$-hyperbolic is subcubic equivalent to the problem of determining whether there is a chordless cycle of length $4$ in a graph. An improved algorithm is also given for both problems, taking advantage of fast rectangular matrix multiplication. In the worst case it runs in $O(n^{3.26})$-time.

In a WDM network, setting up a new wavelength in a fiber requires recalibrating the other wavelengths passing through this fiber. This induces a cost (e.g., time, energy, degradation of QoS) that depends nonlinearly on the number of wavelengths using the fiber. When a set of connection requests must change their optical paths in the network (e.g., during a maintenance operation on a link in the network), the order in which requests are switched affects the total cost of the operation. That is, the reconfiguration of the routing in a WDM network has some cost due to physical layer impairments. We initiate the study of the corresponding optimization problem by modeling the cost of switching a request as a non-linear function depending on the load of the links used by the new lightpath. We prove that determining the optimal rerouting order is NP-complete for a 2-nodes network. We then give general lower and upper bounds on the minimum cost and we identify classes of instances where the problem can be solved in polynomial time. We design heuristics for this problem and analyze their behavior through simulations.

In this paper, we investigate on conceiving reliable fixed broadband wireless networks under outage probability constraints. We introduce a joint model of data routing and bandwidth assignment that minimizes the total renewal fees of licenses. This problem differs from classical capacity planning since the capacity of microwave links is prone to variations and, hence, we must deal with random parameters to guarantee a desirable reliability level of the solution. We introduce a chance-constrained programming approach to tackle this problem and derive integer linear programming (ILP) counterparts. We further propose cutset-based valid inequalities to enhance the performance of ILP solvers. Computational results illustrate the price of reliability and present a comparative study on the performance of the different formulations.

This work introduces a new shared segment protection scheme that ensures both node and link protection in an efficient manner in terms of cost and bandwidth, while taking full advantage of the optical hop endpoints of the primary logical hops (induced by the routing) without adding extra ones for protection. As opposed to the link or path protection schemes, the segment protection scheme has been less studied although it offers an interesting compromise between those two protection schemes, attempting to encompass all their advantages. We investigate two different Shared Segment Protection (SSP) schemes: Basic Shared Segment Protection (BSSP) and Shared Segment Protection with segment Overlap (SSPO), and propose design of 100% single segment protections. In SSPO, we study the extra protection capabilities, node failure and dual link failure survivability, offered by the single 100% segment protection. For both BSSP and SSPO schemes, we propose two novel efficient ILP formulations, based on a column generation mathematical modeling. While (SSPO) offers the advantage over (BSSP) to ensure both node and link protection, it is not necessarily much more costly. Indeed, depending on the network topology and the traffic instances, it can be shown that none of the two SSP schemes dominates the other one. Therefore, the SSPO protection scheme should be favored as it offers more protection, i.e., it adds the node protection to the link protection at the expense of a minor additional cost.

In this paper, we investigate on minimizing the energy consumption of a fixed broadband wireless network through a joint optimization of data routing and radio configuration. Every link holds a set of power-efficient configurations, each of them associating a capacity with its energy cost. The optimization problem involves deciding the network's configuration and flows that minimize the total energy consumption. An exact mathematical formulation of the problem is presented. It relies on a minimum cost multicommodity flow with step increasing cost functions. We then propose a piecewise linear convex function that provides a good approximation of the energy consumption on the links, and present a relaxation of the previous formulation that exploits the convexity of the cost functions. This yields lower bounds on the energy consumption, and finally a heuristic algorithm based on the fractional optimum is employed to produce feasible solutions. Our models are validated through extensive experiments.

The notion of Shared Risk Resource Groups (SRRG) has been introduced to capture survivability issues when a set of resources may fail simultaneously. Applied to Wavelength Division Multiplexing Network (WDM), it expresses that some links and nodes may fail simultaneously. The reliability of a connection therefore depends on the number of SRRGs through which it is routed. Consequently, this number has to be minimized. This problem has been proved NP-complete and hard to approximate in general, even when routing a single request. Some heuristics using shortest paths have already been designed, however the cost (the usual routing cost, not in term of SRRG) was not part of the objective. In this paper we study the problem of minimizing a linear combination of the average number of SRRG per paths and the cost of the routing. The main result of our work is a column generation formulation that allows to solve efficiently the problem of maximizing the reliability of a set of connection requests in MPLS/WDM mesh networks with SRRGs while keeping the cost of the routing low.

SONET/WDM networks using wavelength add-drop multiplexing can be constructed using certain graph decompositions used to form a grooming, consisting of unions of primitive rings. The cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in the graph. The existence of such decompositions with minimum cost, when every pair of sites employs no more than $\frac{1}{6}$ of the wavelength capacity, is determined with a finite number of possible exceptions. Indeed, when the number N of sites satisfies $N \equiv 1 \pmod{3}$, the determination is complete, and when $N \equiv 2 \pmod{3}$, the only value left undetermined is N = 17. When $N \equiv 0 \pmod{3}$, a finite number of values of N remain, the largest being N = 2580. The techniques developed rely heavily on tools from combinatorial design theory.

We address the problem of traffic grooming in WDM rings with all-to-all uniform unitary traffic. We want to minimize the total number of SONET add-drop multiplexers (ADMs) required. We show that this problem corresponds to a partition of the edges of the complete graph into subgraphs, where each subgraph has at most C edges (where C is the grooming ratio) and where the total number of vertices has to be minimized. Using tools of graph and design theory, we optimally solve the problem for practical values and infinite congruence classes of values for a given C, and thus improve and unify all the preceding results. We disprove a conjecture of [A.L. Chiu and E.H. Modiano, 2000] saying that the minimum number of ADMs cannot be achieved with the minimum number of wavelengths and also another conjecture of [J.Q. Hu, 2002].

We consider the problem of finding a lightpath assignment for a given set of communication requests on a multifiber WDM optical network with wavelength translators. Given such a network and w, the number of wavelengths available on each fiber, k, the number of fibers per link, and c, the number of partial wavelength translations available on each node, our problem stands for deciding whether it is possible to find a w-lightpath for each request in the set such that there is no link carrying more that k lightpaths using the same wavelength nor node where more than c wavelength translations take place. Our main theoretical result is the writing of this problem as a particular instance of integral multicommodity flow, hence integrating routing and wavelength assignment in the same model. We then provide three heuristics mainly based upon randomized rounding of fractional multicommodity flow and enhancements that are three different answers to the trade-off between efficiency and tightness of approximation, and discuss their practical performances on both theoretical and real-world instances.

Abstract The de Bruijn digraph B(d, D) has degree d, diameter D, dD vertices, and dD+1 arcs. It is usually defined by words of size D on an alphabet of cardinality d, through a cyclic left‐shift permutation on the words, after which the rightmost symbol is changed. In this paper, we show that any digraph defined on words of a given size, through an arbitrary permutation on the alphabet and an arbitrary permutation on the word indices, is isomorphic to the de Bruijn digraph, provided that this latter permutation is cyclic. We use this result to improve from O(dD+1) to $\Theta(\sqrt{d^{D+1}})$ the number of lenses required for the implementation of B(d, D) by the Optical Transpose Interconnection System proposed by Marsden et al. [Opt Lett 18 (1993), 1083–1085]. © 2002 Wiley Periodicals, Inc.

This study considers the design of a survivable WDM network based on covering the initial network with sub-networks, which are protected independently from each other.

The de Bruijn digraph B(d, D) is usually defined by words of size D on an alphabet of cardinality d, through a cyclic left shift permutation on the words, after which the rightmost symbol is changed. In this paper we show that any digraph defined on words and alphabets of the same size, through an arbitrary permutation on the alphabet and an arbitrary permutation on the word indices, is isomorphic to the de Bruijn, provided that this latter permutation is cyclic. This work is motivated by the next application. It is known that the optical transpose interconnection system from UCSD can implement the de Bruijn interconnections for n nodes, for a fixed d, with O(n) lenses. We show here how to improve this hardware requirement to /spl Theta/(/spl radic/n).