Section: New Results

Graph Algorithms

Participants : Nathann Cohen, David Coudert, Frédéric Giroire, Fatima Zahra Moataz, Benjamin Momège, Nicolas Nisse, Stéphane Pérennes.

Coati is also interested in the algorithmic aspects of Graph Theory. In general we try to find the most efficient algorithms to solve various problems of Graph Theory and telecommunication networks. We use graph theory to model various network problems. We study their complexity and then we investigate the structural properties of graphs that make these problems hard or easy. In particular, we try to find the most efficient algorithms to solve the problems, sometimes focusing on specific graph classes from which the problems are polynomial-time solvable. Many results introduced here are presented in detail in the PhD thesis of F. Z. Moataz [14] .

Graph Hyperbolicity

The Gromov hyperbolicity is an important parameter for analyzing complex networks which expresses how the metric structure of a network looks like a tree (the smaller gap the better). It has recently been used to provide bounds on the expected stretch of greedy-routing algorithms in Internet-like graphs, and for various applications in network security, computational biology, the analysis of graph algorithms, and the classification of complex networks.

Exact Algorithms for Computing the Gromov Hyperbolicity

The best known theoretical algorithm computing this parameter runs in $O\left(n^{3.69}\right)$ time, which is prohibitive for large-scale graphs. In [26] , we propose an algorithm for determining the hyperbolicity of graphs with tens of thousands of nodes. Its running time depends on the distribution of distances and on the actual value of the hyperbolicity. Although its worst case runtime is $O\left(n^4\right)$, it is in practice much faster than previous proposals as observed in our experimentations on benchmark instances. We also propose a heuristic algorithm that can be used on graphs with millions of nodes.

In [37] , we provide a more efficient algorithm: although its worst-case complexity remains in $O\left(n^4\right)$, in practice it is much faster, allowing, for the first time, the computation of the hyperbolicity of graphs with up to $200,000$ nodes. We experimentally show that our new algorithm drastically outperforms the best previously available algorithms, by analyzing a big dataset of real-world networks. We have also used the new algorithm to compute the hyperbolicity of random graphs generated with the Erdös-Renyi model, the Chung-Lu model, and the Configuration Model.

Hyperbolicity of Particular Graph Classes

Topologies for data center networks have been proposed in the literature through various graph classes and operations. A common trait to most existing designs is that they enhance the symmetric properties of the underlying graphs. Indeed, symmetry is a desirable property for interconnection networks because it minimizes congestion problems and it allows each entity to run the same routing protocol. However, despite sharing similarities these topologies all come with their own routing protocol. Recently, generic routing schemes have been introduced which can be implemented for any interconnection networks. The performances of such universal routing schemes are intimately related to the hyperbolicity of the topology. Motivated by the good performances in practice of these new routing schemes, we propose in [56] the first general study of the hyperbolicity of data center interconnection networks. Our findings are disappointingly negative: we prove that the hyperbolicity of most data center topologies scales linearly with their diameter, that it the worst-case possible for hyperbolicity. To obtain these results, we introduce original connection between hyperbolicity and the properties of the endomorphism monoid of a graph. In particular, our results extend to all vertex and edge-transitive graphs. Additional results are obtained for de Bruijn and Kautz graphs, grid-like graphs and networks from the so-called Cayley model.

In [57] , we investigate more specifically on the hyperbolicity of bipartite graphs. More precisely, given a bipartite graph $B=(V_0\cup V_1,E)$ we prove it is enough to consider any one side $V_i$ of the bipartition of $B$ to obtain a close approximate of its hyperbolicity $\delta \left(B\right)$ — up to an additive constant 2. We obtain from this result the sharp bounds $\delta \left(G\right)-1\le \delta \left(L\right(G\left)\right)\le \delta \left(G\right)+1$ and $\delta \left(G\right)-1\le \delta \left(K\right(G\left)\right)\le \delta \left(G\right)+1$ for every graph $G$, with $L\left(G\right)$ and $K\left(G\right)$ being respectively the line graph and the clique graph of $G$. Finally, promising extensions of our techniques to a broader class of intersection graphs are discussed and illustrated with the case of the biclique graph $BK\left(G\right)$, for which we prove $\left(\delta \right(G)-3)/2\le \delta \left(BK\right(G\left)\right)\le \left(\delta \right(G)+3)/2$.

Tree-decompositions

We study the computational complexity of different variants of tree-decompositions. We also study their relationship with various pursuit-evasion games.

Diameter of Minimal Separators in Graphs (structure vs metric in graphs)

In [39] , we establish general relationships between the topological properties of graphs and their metric properties. For this purpose, we upper-bound the diameter of the minimal separators in any graph by a function of their sizes. More precisely, we prove that, in any graph $G$, the diameter of any minimal separator $S$ in $G$ is at most $\lfloor \frac{\ell \left(G\right)}{2}\rfloor ·\left(\right|S|-1)$ where $\ell \left(G\right)$ is the maximum length of an isometric cycle in $G$. We refine this bound in the case of graphs admitting a distance preserving ordering for which we prove that any minimal separator $S$ has diameter at most $2\left(\right|S|-1)$. Our proofs are mainly based on the property that the minimal separators in a graph $G$ are connected in some power of $G$. Our result easily implies that the treelength $tl\left(G\right)$ of any graph $G$ is at most $\lfloor \frac{\ell \left(G\right)}{2}\rfloor$ times its treewidth $tw\left(G\right)$. In addition, we prove that, for any graph $G$ that excludes an apex graph $H$ as a minor, $tw\left(G\right)\le c_H·tl\left(G\right)$ for some constant $c_H$ only depending on $H$. We refine this constant when $G$ has bounded genus. As a consequence, we obtain a very simple $O\left(\ell \right(G\left)\right)$-approximation algorithm for computing the treewidth of $n$-node $m$-edge graphs that exclude an apex graph as a minor in $O\left(nm\right)$-time.

Minimum Size Tree-decompositions

Tree-decompositions are the cornerstone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In [48] , [14] , we consider the problem of computing a tree-decomposition of a graph with width at most $k$ and minimum size. We prove that the problem is NP-complete for any fixed $k\ge 4$ and polynomial for $k\le 2$; for $k=3$, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs.

Non-deterministic Graph Searching in Trees

Non-deterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given $q\ge 0$, the $q$-limited search number, $s_q\left(G\right)$, of a graph $G$ is the smallest number of searchers required to capture an invisible fugitive in $G$, when the searchers are allowed to know the position of the fugitive at most $q$ times. The search parameter $s_0\left(G\right)$ corresponds to the pathwidth of a graph $G$, and $s_{\infty }\left(G\right)$ to its treewidth. Determining $s_q\left(G\right)$ is NP-complete for any fixed $q\ge 0$ in general graphs and $s_0\left(T\right)$ can be computed in linear time in trees, however the complexity of the problem on trees has been unknown for any $q>0$. We introduce in [16] a new variant of graph searching called restricted non-deterministic. The corresponding parameter is denoted by $r{s}_q$ and is shown to be equal to the non-deterministic graph searching parameter $s_q$ for $q=0,1$, and at most twice $s_q$ for any $q\ge 2$ (for any graph $G$). Our main result is a polynomial time algorithm that computes $r{s}_q\left(T\right)$ for any tree $T$ and any $q\ge 0$. This provides a 2-approximation of $s_q\left(T\right)$ for any tree $T$ , and shows that the decision problem associated to $s_1$ is polynomial in the class of trees. Our proofs are based on a new decomposition technique for trees which might be of independent interest.

$k$-Chordal Graphs: from Cops and Robber to Compact Routing via Treewidth

Cops and robber games, introduced by Winkler and Nowakowski and independently defined by Quilliot, concern a team of cops that must capture a robber moving in a graph. We consider in [34] the class of $k$-chordal graphs, i.e., graphs with no induced (chordless) cycle of length greater than $k$, $k\ge 3$. We prove that $k-1$ cops are always sufficient to capture a robber in $k$-chordal graphs. This leads us to our main result, a new structural decomposition for a graph class including $k$-chordal graphs. We present a polynomial-time algorithm that, given a graph $G$ and $k\ge 3$, either returns an induced cycle larger than $k$ in $G$, or computes a tree-decomposition of $G$, each bag of which contains a dominating path with at most $k-1$ vertices. This allows us to prove that any $k$-chordal graph with maximum degree $\Delta$ has treewidth at most $\left(k-1\right)\left(\Delta -1\right)+2$, improving the $O\left(\Delta \right(\Delta -1\left)k-3\right)$ bound of Bodlaender and Thilikos (1997). Moreover, any graph admitting such a tree-decomposition has small hyperbolicity. As an application, for any $n$-vertex graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size $O(klog\Delta +logn)$ bits and achieving an additive stretch of $O(klog\Delta )$. As far as we know, this is the first routing scheme with $O(klog\Delta +logn)$-routing tables and small additive stretch for $k$-chordal graphs.

Connected Surveillance Game

The surveillance game [68] models the problem of web-page prefetching as a pursuit evasion game played on a graph. This two-player game is played turn-by-turn. The first player, called the observer, can mark a fixed amount of vertices at each turn. The second one controls a surfer that stands at vertices of the graph and can slide along edges. The surfer starts at some initially marked vertex of the graph, its objective is to reach an unmarked node before all nodes of the graph are marked. The surveillance number $sn\left(G\right)$ of a graph $G$ is the minimum amount of nodes that the observer has to mark at each turn ensuring it wins against any surfer in $G$. Fomin et al. also defined the connected surveillance game where the observer must ensure that marked nodes always induce a connected subgraph. They ask what is the cost of connectivity, i.e., is there a constant $c>0$ such that the ratio between the connected surveillance number $csn\left(G\right)$ and $sn\left(G\right)$ is at most $c$ for any graph $G$. It is straightforward to show that $csn\left(G\right)\le \Delta sn\left(G\right)$ for any graph $G$ with maximum degree $\Delta$. Moreover, it has been shown that there are graphs $G$ for which $csn\left(G\right)=sn\left(G\right)+1$. In [30] , we investigate the question of the cost of the connectivity. We first provide new non-trivial upper and lower bounds for the cost of connectivity in the surveillance game. More precisely, we present a family of graphs $G$ such that $csn\left(G\right)>sn\left(G\right)+1$. Moreover, we prove that $csn\left(G\right)\le \sqrt{sn\left(G\right)n}$ for any $n$-node graph $G$. While the gap between these bounds remains huge, it seems difficult to reduce it. We then define the online surveillance game where the observer has no a priori knowledge of the graph topology and discovers it little-by-little. This variant, which fits better the prefetching motivation, is a restriction of the connected variant. Unfortunately, we show that no algorithm for solving the online surveillance game has competitive ratio better than $\Omega \left(\Delta \right)$. That is, while interesting, this variant does not help to obtain better upper bounds for the connected variant. We finally answer an open question [68] by proving that deciding if the surveillance number of a digraph with maximum degree 6 is at most 2 is NP-hard.

Distributed Algorithms

Allowing each Node to Communicate only once in a Distributed System: Shared Whiteboard Models

In [21] we study distributed algorithms on massive graphs where links represent a particular relationship between nodes (for instance, nodes may represent phone numbers and links may indicate telephone calls). Since such graphs are massive they need to be processed in a distributed way. When computing graph-theoretic properties, nodes become natural units for distributed computation. Links do not necessarily represent communication channels between the computing units and therefore do not restrict the communication flow. Our goal is to model and analyze the computational power of such distributed systems where one computing unit is assigned to each node. Communication takes place on a whiteboard where each node is allowed to write at most one message. Every node can read the contents of the whiteboard and, when activated, can write one small message based on its local knowledge. When the protocol terminates its output is computed from the final contents of the whiteboard. We describe four synchronization models for accessing the whiteboard. We show that message size and synchronization power constitute two orthogonal hierarchies for these systems.We exhibit problems that separate these models, i.e., that can be solved in one model but not in a weaker one, even with increased message size. These problems are related to maximal independent set and connectivity. We also exhibit problems that require a given message size independently of the synchronization model.

Computing on Rings by Oblivious Robots: a Unified Approach for Different Tasks

A set of autonomous robots have to collaborate in order to accomplish a common task in a ring-topology where neither nodes nor edges are labeled (that is, the ring is anonymous). In [36] , we present a unified approach to solve three important problems: the exclusive perpetual exploration, the exclusive perpetual clearing, and the gathering problems. In the first problem, each robot aims at visiting each node infinitely often while avoiding that two robots occupy a same node (exclusivity property); in exclusive perpetual clearing (also known as searching), the team of robots aims at clearing the whole ring infinitely often (an edge is cleared if it is traversed by a robot or if both its endpoints are occupied); and in the gathering problem, all robots must eventually occupy the same node. We investigate these tasks in the Look-Compute-Move model where the robots cannot communicate but can perceive the positions of other robots. Each robot is equipped with visibility sensors and motion actuators, and it operates in asynchronous cycles. In each cycle, a robot takes a snapshot of the current global configuration (Look), then, based on the perceived configuration, takes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case it eventually moves to this neighbor (Move). Moreover, robots are endowed with very weak capabilities. Namely, they are anonymous, asynchronous, oblivious, uniform (execute the same algorithm) and have no common sense of orientation. In this setting, we devise algorithms that, starting from an exclusive and rigid (i.e. aperiodic and asymmetric) configuration, solve the three above problems in anonymous ring-topologies.

Miscellaneous

Finding Paths in Grids with Forbidden Transitions

A transition in a graph is a pair of adjacent edges. Given a graph $G=(V,E)$, a set of forbidden transitions $F\subseteq E\times E$ and two vertices $s,t\in V$ , we study in [64] , [45] , [46] , [14] the problem of finding a path from $s$ to $t$ which uses none of the forbidden transitions of $F$. This means that it is forbidden for the path to consecutively use two edges forming a pair in $F$. The study of this problem is motivated by routing in road networks in which forbidden transitions are associated to prohibited turns as well as routing in optical networks with asymmetric nodes, which are nodes where a signal on an ingress port can only reach a subset of egress ports. If the path is not required to be elementary, the problem can be solved in polynomial time. On the other side, if the path has to be elementary, the problem is known to be NP-complete in general graphs [69] . In [45] , we study the problem of finding an elementary path avoiding forbidden transitions in planar graphs. We prove that the problem is NP-complete in planar graphs and particularly in grids. In addition, we show that the problem can be solved in polynomial time in graphs with bounded treewidth. More precisely, we show that there is an algorithm which solves the problem in time $O\left(\right(3\Delta (k+1))2k+4n))$ in $n$-node graphs with treewidth at most $k$ and maximum degree $\Delta$.