Section: New Results

Graph and Combinatorial Algorithms

Rainbow matchings in hypergraphs

A rainbow matching for (not necessarily distinct) sets $F_1,...,F_k$ of hypergraph edges is a matching consisting of $k$ edges, one from each $F_i$. In [8] , we give some order to the multitude of conjectures that relate to this concept, as well as introduce some new conjectures. We also present some partial results on one of these conjectures, that seems central among them – the so-called Ryser-Brualdi-Stein conjecture.

A graph formulation of the union-closed sets conjecture

In 1979, Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. In [7] , we show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices, each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it also holds for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs.

Cops-and-robber games on $k$-chordal graphs

The cops-and-robber games, introduced by Winkler and Nowakowski (in Discrete Math. 43, 1983) and independently defined by Quilliot (in J. Comb. Theory, Ser. B 38, 1985), concern a team of cops that must capture a robber moving in a graph. In [20] , we consider 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 (Discrete Appl. Math. 79, 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.

Distinguishing views in symmetric networks

The view of a node in a port-labeled network is an infinite tree encoding all walks in the network originating from this node. In [16] , we prove that for any integers $n\ge D\ge 1$, there exists a port-labeled network with at most $n$ nodes and diameter at most $D$, which contains a pair of nodes whose (infinite) views are different, but whose views truncated to depth $\Omega (Dlog(n/D\left)\right)$ are identical.

Vertex elimination orderings for hereditary graph classes

In [3] , we provide a general method to prove the existence and compute efficiently elimination orderings in graphs. This method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, its additional properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vušković. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we consider can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem.

Fast collaborative graph exploration

In [14] , we study the following scenario of online graph exploration. A team of $k$ agents is initially located at a distinguished vertex $r$ of an undirected graph. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. As our main result, we provide the first strategy which performs exploration of a graph with $n$ vertices at a distance of at most $D$ from $r$ in time $O\left(D\right)$, using a team of agents of polynomial size $k=D{n}^{1+ϵ}<n^{2+ϵ}$, for any $ϵ>0$. Our strategy works in the local communication model, in which agents can only exchange information when located at a vertex, without knowledge of global parameters such as $n$ or $D$.

We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large $k$, in both the local and the global communication model.

Position discovery for a system of bouncing robots

In [11] , we consider a scenario in which a collection of $n$ anonymous mobile robots is deployed on a unit-perimeter ring or a unit-length line segment. Every robot starts moving at constant speed, and bounces each time it meets any other robot or segment endpoint, changing its walk direction. We study the problem of position discovery, in which the task of each robot is to detect the presence and the initial positions of all other robots. The robots cannot communicate or perceive information about the environment in any way other than by bouncing nor they have control over their walks which are determined by their initial positions and their starting directions. Each robot has a clock allowing it to observe the times of its bounces. We give complete characterizations of all initial configurations for both the ring and the segment in which no position detection algorithm exists and we design optimal position detection algorithms for all feasible configurations.

Rendezvous of mobile agents in edge-weighted networks

In [15] , we introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time $T_{OPT}$ in the offline scenario in which the agents have complete knowledge about each others' speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time $O\left(n{T}_{OPT}\right)$ in a $n$-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to $\Theta \left(T_{OPT}\right)$ when the agents are allowed to exchange $\Theta \left(n\right)$ bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time $\Theta \left(T_{OPT}\right)$.

Monitoring a graph using faulty mobile robots

In the scenario studied in [27] , a team of $k$ mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots perpetually move along the domain, represented by all points belonging to the graph edges, not exceeding their maximal speed. The robots need to patrol the graph by regularly visiting all points of the domain. Here, we consider a team of robots (patrolmen), at most $f$ of which may be unreliable, i.e. they fail to comply with their patrolling duties.

What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolmen? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by $I_k^f\left(G\right)$ the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing $I_k^f\left(G\right)$.

We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study the case of general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness $I_k^f\left(G\right)=(f+1)\left|E\right|/k$, where $\left|E\right|$ is the sum of the lengths of the edges of $G$. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs — a known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs.

Limit behavior of the rotor-router system

The rotor-router model, also called the Propp machine, was introduced as a deterministic alternative to the random walk. In this model, a group of identical tokens are initially placed at nodes of the graph. Each node maintains a cyclic ordering of the outgoing arcs, and during consecutive turns the tokens are propagated along arcs chosen according to this ordering in round-robin fashion. The behavior of the model is fully deterministic. Yanovski et al. (Algorithmica, 2003) proved that a single rotor-router walk on any graph with $m$ edges and diameter $D$ stabilizes to a traversal of an Eulerian circuit on the set of all $2m$ directed arcs on the edge set of the graph, and that such periodic behaviour of the system is achieved after an initial transient phase of at most $2mD$ steps.

The case of multiple parallel rotor-routers was studied experimentally, leading Yanovski et al. to the experimental observation that a system of $k>1$ parallel walks also stabilizes with a period of length at most $2m$ steps. In our work [26] we disprove this observation, showing that the period of parallel rotor-router walks can in fact, be superpolynomial in the size of graph. On the positive side, we provide a characterization of the periodic behavior of parallel router walks, in terms of a structural property of stable states called a subcycle decomposition. This property provides us the tools to efficiently detect whether a given system configuration corresponds to the transient or to the limit behavior of the system. Moreover, we provide polynomial upper bounds of $O(m^4{D}^2+mDlogk)$ and $O\left(m^5{k}^2\right)$ on the number of steps it takes for the system to stabilize. Thus, we are able to predict any future behavior of the system using an algorithm that takes polynomial time and space. In addition, we show that there exists a separation between the stabilization time of the single-walk and multiple-walk rotor-router systems, and that for some graphs the latter can be asymptotically larger even for the case of $k=2$ walks.