Section: New Results

Graph and Combinatorial Algorithms

Random Walks with Multiple Step Lengths

In nature, search processes that use randomly oriented steps of different lengths have been observed at both the microscopic and the macroscopic scales. Physicists have analyzed in depth two such processes on grid topologies: Intermittent Search, which uses two step lengths, and Lévy Walk, which uses many. Taking a computational perspective, in [26] we consider the number of distinct step lengths $k$ as a complexity measure of the considered process. Our goal is to understand what is the optimal achievable time needed to cover the whole terrain, for any given value of $k$. Attention is restricted to dimension one, since on higher dimensions, the simple random walk already displays a quasi linear cover time.

We say $X$ is a $k$-intermittent search on the one dimensional $n$-node cycle if there exists a probability distribution $𝐩={\left(p_i\right)}_{i=1}^k$, and integers $L_1,L_2,...,L_k$, such that on each step $X$ makes a jump $\pm L_i$ with probability $p_i$, where the direction of the jump ($+$ or $-$) is chosen independently with probability $1/2$. When performing a jump of length $L_i$, the process consumes time $L_i,$ and is only considered to visit the last point reached by the jump (and not any other intermediate nodes). This assumption is consistent with biological evidence, in which entities do not search while moving ballistically. We provide upper and lower bounds for the cover time achievable by $k$-intermittent searches for any integer $k$. In particular, we prove that in order to reduce the cover time $\Theta \left(n^2\right)$ of a simple random walk to $\tilde{\Theta }\left(n\right)$, roughly $\frac{logn}{loglogn}$ step lengths are both necessary and sufficient, and we provide an example where the lengths form an exponential sequence.

In addition, inspired by the notion of intermittent search, we introduce the Walk or Probe problem, which can be defined with respect to arbitrary graphs. Here, it is assumed that querying (probing) a node takes significantly more time than moving to a random neighbor. Hence, to efficiently probe all nodes, the goal is to balance the time spent walking randomly and the time spent probing. We provide preliminary results for connected graphs and regular graphs.

Searching a Tree with Permanently Noisy Advice

In [16], we consider a search problem on trees using unreliable guiding instructions. Specifically, an agent starts a search at the root of a tree aiming to find a treasure hidden at one of the nodes by an adversary. Each visited node holds information, called advice, regarding the most promising neighbor to continue the search. However, the memory holding this information may be unreliable. Modeling this scenario, we focus on a probabilistic setting. That is, the advice at a node is a pointer to one of its neighbors. With probability $q$ each node is faulty, independently of other nodes, in which case its advice points at an arbitrary neighbor, chosen uniformly at random. Otherwise, the node is sound and points at the correct neighbor. Crucially, the advice is permanent, in the sense that querying a node several times would yield the same answer. We evaluate efficiency by two measures: The move complexity denotes the expected number of edge traversals, and the query complexity denotes the expected number of queries.

Let $\Delta$ denote the maximal degree. Roughly speaking, the main message of this paper is that a phase transition occurs when the noise parameter $q$ is roughly $1/\sqrt{\Delta }$. More precisely, we prove that above the threshold, every search algorithm has query complexity (and move complexity) which is both exponential in the depth $d$ of the treasure and polynomial in the number of nodes $n$. Conversely, below the threshold, there exists an algorithm with move complexity $O\left(d\sqrt{\Delta }\right)$, and an algorithm with query complexity $O(\sqrt{\Delta }log\Delta {log}^{2}n)$. Moreover, for the case of regular trees, we obtain an algorithm with query complexity $O(\sqrt{\Delta }lognloglogn)$. For $q$ that is below but close to the threshold, the bound for the move complexity is tight, and the bounds for the query complexity are not far from the lower bound of $\Omega \left(\sqrt{\Delta }{log}_{\Delta }n\right)$.

In addition, we also consider a semi-adversarial variant, in which faulty nodes are still chosen at random, but an adversary chooses (beforehand) the advice of such nodes. For this variant, the threshold for efficient moving algorithms happens when the noise parameter is roughly $1/\Delta$. In fact, above this threshold a simple protocol that follows each advice with a fixed probability already achieves optimal move complexity.

Patterns on 3 vertices

In [31] we deal with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals.

We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time.

Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.

The Dependent Doors Problem: An Investigation into Sequential Decisions without Feedback

In [13], we introduce the dependent doors problem as an abstraction for situations in which one must perform a sequence of dependent decisions, without receiving feedback information on the effectiveness of previously made actions. Informally, the problem considers a set of $d$ doors that are initially closed, and the aim is to open all of them as fast as possible. To open a door, the algorithm knocks on it and it might open or not according to some probability distribution. This distribution may depend on which other doors are currently open, as well as on which other doors were open during each of the previous knocks on that door. The algorithm aims to minimize the expected time until all doors open. Crucially, it must act at any time without knowing whether or which other doors have already opened. In this work, we focus on scenarios where dependencies between doors are both positively correlated and acyclic.

The fundamental distribution of a door describes the probability it opens in the best of conditions (with respect to other doors being open or closed). We show that if in two configurations of $d$ doors corresponding doors share the same fundamental distribution, then these configurations have the same optimal running time up to a universal constant, no matter what are the dependencies between doors and what are the distributions. We also identify algorithms that are optimal up to a universal constant factor. For the case in which all doors share the same fundamental distribution we additionally provide a simpler algorithm, and a formula to calculate its running time. We furthermore analyse the price of lacking feedback for several configurations governed by standard fundamental distributions. In particular, we show that the price is logarithmic in $d$ for memoryless doors, but can potentially grow to be linear in $d$ for other distributions.

We then turn our attention to investigate precise bounds. Even for the case of two doors, identifying the optimal sequence is an intriguing combinatorial question. Here, we study the case of two cascading memoryless doors. That is, the first door opens on each knock independently with probability $p_1$. The second door can only open if the first door is open, in which case it will open on each knock independently with probability $p_2$. We solve this problem almost completely by identifying algorithms that are optimal up to an additive term of 1.

Finding maximum cliques in disk and unit ball graphs

In an intersection graph, the vertices are geometric objects with an edge between any pair of intersecting objects. Intersection graphs have been studied for many different families of objects due to their practical applications and their rich structural properties. Among the most studied ones are disk graphs, which are intersection graphs of closed disks in the plane, and their special case, unit disk graphs, where all the radii are equal. Their applications range from sensor networks to map labeling, and many standard optimization problems have been studied on disk graphs. Most of the hard optimization and decision problems remain NP-hard on disk graphs and even unit disk graphs. For instance, disk graphs contain planar graphs on which several of those problems are intractable.

The complexity of Maximum Clique on general disk graphs is a notorious open question in computational geometry. On the one hand, no polynomial-time algorithm is known, even when the geometric representation is given. On the other hand, the NP-hardness of the problem has not been established, even when only the graph is given as input.

Recently, Bonnet et al. showed that the disjoint union of two odd cycles is not the complement of a disk graph. From this result, they obtained a subexponential algorithm running in time $2^{\tilde{O}\left(n^{2/3}\right)}$ for Maximum Clique on disk graphs, based on a win-win approach. They also got a QPTAS by calling a PTAS for Maximum Independent Set on graphs with sublinear odd cycle packing number due to Bock et al., or branching on a low-degree vertex.

In [17], our main contributions are twofold. The first is a randomized EPTAS (Efficient Polynomial-Time Approximation Scheme, that is, a PTAS in time $f\left(\epsilon \right)n^{O\left(1\right)}$) for Maximum Independent Set on graphs of $𝒳(d,\beta ,1)$. The class $𝒳(d,\beta ,1)$ denotes the class of graphs whose neighborhood hypergraph has VC-dimension at most $d$, independence number at least $\beta n$, and no disjoint union of two odd cycles as an induced subgraph. Using the forbidden induced subgraph result of Bonnet et al., it is then easy to reduce Maximum Clique on disk graphs to Maximum Independent Set on $𝒳(4,\beta ,1)$ for some constant $\beta$. We therefore obtain a randomized EPTAS (and a PTAS) for Maximum Clique on disk graphs, settling almost (The NP-hardness, ruling out a 1-approximation, is still to show.) completely the approximability of this problem.

The second contribution is to show the same forbidden induced subgraph for unit ball graphs as the one obtained for disk graphs : their complement cannot have a disjoint union of two odd cycles as an induced subgraph. The proofs are radically different and the classes are incomparable. So the fact that the same obstruction applies for disk graphs and unit ball graphs might be somewhat accidental. And again we therefore obtain a randomized EPTAS in time $2^{\tilde{O}\left(1/{\epsilon }^3\right)}{n}^{O\left(1\right)}$ for Maximum Clique on unit ball graphs, even without the geometric representation.

Before that result, the best approximation factor was 2.553, due to Afshani and Chan. In particular, even getting a 2-approximation algorithm (as for disk graphs) was open.

Finally we show that such an approximation scheme, even in subexponential time, is unlikely for ball graphs (that is, 3-dimensional disk graphs with arbitrary radii), and unit 4-dimensional disk graphs. Our lower bounds also imply NP-hardness. To the best of our knowledge, the NP-hardness of Maximum Clique on unit $d$-dimensional disk graphs was only known when $d$ is superconstant ($d=\Omega (logn)$).

$\delta$-hyperbolicity

In [19], we show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O\left(1\right)$ time an estimate $\widehat{e}\left(v\right)$ of its eccentricity $ec{c}_G\left(v\right)$ such that $ec{c}_G\left(v\right)\le \widehat{e}\left(v\right)\le ec{c}_G\left(v\right)+c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ec{c}_G\left(v\right)\le ec{c}_T\left(v\right)\le ec{c}_G\left(v\right)+c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c^{\text{'}}\delta$ can be computed in $O\left(\right|V{|}^2{log}^2\left|V\right|)$ time, where $c^{\text{'}}<c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

Graph searches and geometric convexities in graphs

In an attempt to understand graph searching on cocomparability graphs has been so successful, one quickly notices that the orderings produced by these traversals are precisely words of some antimatroids or convex geometries. The notion of antimatroids and convex geometries have appeared in the literature under various settings; in this work, we focus on the graph searching setting, where we discuss some known geometries on cocomparability graphs, and then present new structural properties on AT-free graphs in the hope of exploring whether the algorithms on cocomparability graphs can be lifted to this larger graph class. A first version of this work in collaboration with Feodor Dragan and Lalla Mouatadib was presented at ICGT Lyon, july 2018.