Ioan Todinca

Université d'Orléans | UO · Département d'Informatique

We study the design of energy-efficient algorithms for the LOCAL and CONGEST models. Specifically, as a measure of complexity, we consider the maximum, taken over all the edges, or over all the nodes, of the number of rounds at which an edge, or a node, is active in the algorithm. We first show that every Turing-computable problem has a CONGEST alg...

Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class...

We consider the Trivially Perfect Editing problem, where one is given an undirected graph \(G = (V,E)\) and a parameter \(k \in {\mathbb {N}}\) and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing e...

During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant numb...

We show that if the edges or vertices of an undirected graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is upper-bounded by a function of $k$. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph $G$ and a set of $k$ pairs of vertices called \emph{terminals}, asks whether $G$ c...

We consider edge modification problems towards block and strictly chordal graphs, where one is given an undirected graph $G = (V,E)$ and an integer $k \in \mathbb{N}$ and seeks to edit (add or delete) at most $k$ edges from $G$ to obtain a block graph or a strictly chordal graph. The completion and deletion variants of these problems are defined si...

Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to each process in the system by a non-trustable oracle, and the processes are in charge of verifying these cert...

Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to each process in the system by a non-trustable oracle, and the processes are in charge of verifying these cert...

We consider the Strictly Chordal Editing problem, where one is given an undirected graph G = (V,E) and a parameter k ∈ ℕ and seeks to edit (add or delete) at most k edges from G to obtain a strictly chordal graph. Problems Strictly Chordal Completion and Strictly Chordal Deletion are defined similarly, by only allowing edge additions for the former...

We consider the Trivially Perfect Editing problem, where one is given an undirected graph G = (V,E) and a parameter k ∈ ℕ and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge...

Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACM-SIAM symposium on discrete algorithms (SODA), pp 1096–115, 2020. https://doi.org/10.1137/1.9781611975994.67) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar),...

We study the role of randomness in the broadcast congested clique model. This is a message-passing model of distributed computation where the nodes of a network know their local neighborhoods and they broadcast, in synchronous rounds, messages that are visible to every other node. This works aims to separate three different settings: deterministic...

Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class...

Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a \emph{distributed interactive proof} for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a di...

In a distributed locally-checkable proof, we are interested in checking the legality of a given network configuration with respect to some Boolean predicate. To do so, the network enlists the help of a prover—a computationally-unbounded oracle that aims at convincing the network that its state is legal, by providing the nodes with certificates that...

Let \(\mathcal {P}(G,X)\) be a property associating a boolean value to each pair (G, X) where G is a graph and X is a vertex subset. Assume that \(\mathcal {P}\) is expressible in counting monadic second order logic (CMSO) and let t be an integer constant. We consider the following optimization problem: given an input graph \(G=(V,E)\), find subset...

In this paper we study the reconstruction problem in the congested clique model. In the reconstruction problem nodes are asked to recover all the edges of the input graph G. Formally, given a class of graphs \(\mathcal G\), the problem is defined as follows: if \(G \notin {\mathcal G}\), then every node must reject; on the other hand, if \(G \in {\...

In the standard CONGEST model for distributed network computing, it is known that "global" tasks such as minimum spanning tree, diameter, and all-pairs shortest paths, consume large bandwidth, for their running-time is $\Omega(\mbox{poly}(n))$ rounds in $n$-node networks with constant diameter. Surprisingly, "local" tasks such as detecting the pres...

The congested clique model is a message-passing model of distributed computation where the underlying communication network is the complete graph of $n$ nodes. In this paper we consider the situation where the joint input to the nodes is an $n$-node labeled graph $G$, i.e., the local input received by each node is the indicator function of its neig...

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (\({\text {vc}}\)) and modular width (\({\text {mw}}\)). We prove that for any graph, the number of its minimal separators is \({\mathcal {O}}^*(3^{{\text {vc}}})\) and \({\mathcal {O}}^*(1....

After the number of vertices, Vertex Cover Number is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover Number. Here we consider the TREEWIDTH and PATHWIDTH problems parameterized by k, the si...

We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing wheth...

In many graph problems, like Longest Induced Path, Maximum Induced Forest, etc., we are given as input a graph G and the goal is to compute a largest induced subgraph G[F], of treewidth at most a constant t, and satisfying some property \(\mathcal {P}\). Fomin et al. [12] proved that this generic problem is polynomial on the class of graphs \({\mat...

We present deterministic constant-round protocols for the graph connectivity problem in the model where each of the n nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Specifi...

Fomin and Villanger (STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant $t$, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let $\Gpoly$ be th...

Fomin and Villanger ([14], STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let \(\mathcal...

We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing wheth...

We present a deterministic constant-round protocol for the graph connectivity problem in the model where each of the $n$ nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Spec...

We study the multiparty communication model where players are the nodes of a network and each of these players knows his/her own identifier together with the identifiers of his/her neighbors. The players simultaneously send a unique message to a referee who must decide a graph property. The goal of this article is to separate, from the point of vie...

An injective coloring of a graph is a vertex labeling such that two vertices sharing a common neighbor get different labels. In this work we introduce and study what we call additive colorings. An injective coloring \(c:V(G)\rightarrow \mathbb {Z}\) of a graph \(G\) is an additive coloring if for every \(uv, vw\) in \(E(G)\), \(c(u)+c(w)\ne 2c(v)\)...

Consider an undirected graph GG and a subgraph HH of GG, on the same vertex set. The qq-backbone chromatic numberBBCq(G,H) is the minimum kk such that GG can be properly coloured with colours from {1,…,k}{1,…,k}, and moreover for each edge of HH, the colours of its ends differ by at least qq. In this paper we focus on the case when GG is planar and...

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover ($\operatorname{vc}$) and modular width ($\operatorname{mw}$). We prove that for any graph, the number of minimal separators is $\mathcal{O}^*(3^{\operatorname{vc}})$ and $\mathcal{O}^*(1.61...

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at m...

We study the problem of predicting the state of a vertex in automata networks, where the state at each site is given by the majority function over its neighborhood. We show that for networks with maximum degree greater than 5 the problem is P-Complete, simulating a monotone Boolean circuit. Then, we show that the problem for networks with no vertex...

After the number of vertices, Vertex Cover is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover. Here we consider the TREEWIDTH and PATHWIDTH problems parameterized by k, the size of a minimu...

Given a graph GG with tree-width ω(G)ω(G), branch-width β(G)β(G), and side size of the largest square grid-minor θ(G)θ(G), it is known that θ(G)≤β(G)≤ω(G)+1≤32β(G). In this paper, we introduce another approach to bound the side size of the largest square grid-minor specifically for planar graphs. The approach is based on measuring the distances bet...

In this paper we ask which properties of a distributed network can be computed from a little amount of local information provided by its nodes. The distributed model we consider is a restriction of the classical CONGEST (distributed) model and it is close to the simultaneous messages (communication complexity) model defined by Babai, Kimmel and Lok...

The minimal interval completion problem consists in adding edges to an arbitrary graph so that the resulting graph is an interval graph; the objective is to add an inclusion minimal set of edges, which means that no proper subset of the added edges can result in an interval graph when added to the original graph. We give an O(n 2) time algorithm to...

The Capacitated Dominating Set problem is the problem of finding a dominating set of minimum cardinality where each vertex has been assigned a bound on the number of vertices it has capacity to dominate. Cygan et al. showed in 2009 that this problem can be solved in O(n3 m ((n) || (n/3)))O(n^3 m {{n} \choose {n/3}}) or in O *(1.89 n ) time using ma...

The minimal interval completion problem consists in adding edges to an arbitrary graph so that the resulting graph is an interval graph; the objective is to add an inclusion minimal set of edges, which means that no proper subset of the added edges can result in an interval graph when added to the original graph. We give an O(n 2)-time algorithm to...

The minimum dominating set problem remains NP-hard when restricted to any of the following graph classes: c-dense graphs, chordal graphs, 4-chordal graphs, weakly chordal graphs, and circle graphs. Developing and using a general approach, for each of these graph classes we present an exponential time algorithm solving the minimum dominating set pro...

Given an arbitrary graph G and a number k, it is well-known by a result of Seymour and Thomas that G has treewidth strictly larger than k if and only if it has a bramble of order k + 2. Brambles are used in combinatorics as certificates proving that the treewidth of a graph is large. From an algorithmic point of view there are several algorithms com...

The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G. We prove in this paper that the pathwidth problem is NP-hard for particular subclasses of chordal graphs, and we deduce that the problem remains hard for weighted trees. We also discuss subclasses of chordal graphs for which the problem is...

Minimal triangulations and potential maximal cliques are the main ingredients for a number of polynomial time algorithms on different graph classes computing the treewidth of a graph. Potential maximal cliques are also the main engine of the fastest so far \(\mathcal{O}\)(1.9601n )-time exact treewidth algorithm. Based on the recent results of Mazo...

Given an arbitrary graph G=(V,E) and an interval graph H=(V,F) with E⊆F we say that H is an interval completion of G. The graph H is called a minimal interval completion of G if, for any sandwich graph H′=(V,F′) with E⊆F′⊂F, H′ is not an interval graph. In this paper we give a O(nm) time algorithm computing a minimal interval completion of an arbit...

We present a polynomial time algorithm to compute a minimum (weight) feedback vertex set for AT-free graphs, and extending this approach we obtain a polynomial time algorithm for graphs of bounded asteroidal number.

We investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter...

We investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter...

We show that the treewidth and the minimum fill-in of an n-vertex graph can be com- puted in time O(1.8899n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087n) minimal separators and O(1.8135n) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be red...

Given a graph G, the graph Gl has the same vertex set and two vertices are adjacent in Gl if and only if they are at distance at most l in G. The l-coloring problem consists in finding an optimal vertex coloring of the graph Gl, where G is the input graph. We show that, for any fixed value of l, the l-coloring problem is polynomial when restricted...

The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G. Although pathwidth is a well-known and well-studied graph parameter, there are extremely few graph classes for which pathwidh is known to be tractable in polynomial time. We give in this paper an O(n2)-time algorithm computing the pathwidth...

Minimal interval completions of graphs are central in understanding two important and widely studied graph parameters: profile and pathwidth. Such understanding seems necessary to be able to attack the problem of computing these parameters. An interval completion of a given graph is an interval supergraph of it on the same vertex set, obtained by a...

Given an arbitrary graph G = (V,E) and an interval graph H = (V,F) with E F we say that H is an interval completion of G. The graph H is called a minimal interval completion of G if, for any sandwich graph H0 = (V,F 0) with E F 0 F, H0 is not an interval graph. In this paper we give a O(nm) time algorithm computing a minimal interval completion of...

Given an arbitrary graph G = (V, E) and a proper interval graph H = (V, F) with E C F we say that H is a proper interval completion of G. The graph H is called a minimal proper interval completion of G if, for any sandwich graph H, = (V, 171) with E subset of F' subset of F, H' is not a proper interval graph. In this paper we give a Omicron(n + in)...

We study the problem of adding edges to an arbitrary graph so that the resulting graph is an interval graph. Our objective is to add an inclusion minimal set of edges, which means that no proper subset of the added edges can result in an interval graph when added to the original graph. This problem is closely related to the problem of adding an inc...

We show that for a graph G on n vertices its treewidth and minimum fill-in can be computed roughly in 1.9601n time. Our result is based on a combinatorial proof that the number of minimal separators in a graph is \(\mathcal O(n \cdot 1.7087^n)\) and that the number of potential maximal cliques s is \(\mathcal O(n^4 \cdot 1.9601^n)\).

We introduce a natural heuristic for approximating the treewidth of graphs. We prove that this heuristic gives a constant factor approximation for the treewidth of graphs with bounded asteroidal number. Using a different technique, we give a approximation algorithm for the treewidth of arbitrary graphs, where k is the treewidth of the input graph.

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.

Using the specific structure of the minimal separators of AT-free graphs, we give a polynomial time algorithm that computes a triangulation whose width is no more than twice the treewidth of the input graph.

We give a polynomial time algorithm to compute a minimum (weight) feedback vertex set for AT-free graphs, and extending this approach we obtain a polynomial time algorithm for graphs of bounded asteroidal number. We also present an O(nm 2) algorithm to compute a longest induced path in AT-free graphs.

Given a graph G, the graph G l has the same vertex set and two vertices are adjacent in G l if and only if they are at distance at most l in G. The l-coloring problem consists in finding an optimal vertex coloring of the graph G l , where G the input graph. We show that, for any fixed value of l, the l-coloring problem is polynomial when restricted...

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.

We introduce a natural heuristic for approximating the treewidth of graphs. We prove that this heuristic gives a constant factor approximation for the treewidth of graphs with bounded asteroidal number. Using a different technique, we give a $O(\log k)$ approximation algorithm for the treewidth of arbitrary graphs, where $k$ is the treewidth of the...

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which p...

Abstract A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum 5ll-in are polynomially tractable for these graphs. We show here that the potential maximal...

Using the specific structure of the minimal separators of AT-free graphs, we give a polynomial time algorithm that computes a triangulation whose width is no more than twice the treewidth of the input graph.

A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We show here that the potential maximal cliques...

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum ll-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which pol...

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. Finally we show how to compute in polynomial time t...

We introduce the notion of ipotential maximal cliquej of a graph and we use it for computing the treewidth and the minimum ll-in of graphs for which the the potential maximal cliques can be listed in polynomial time. Finally we show how to compute the potential maximal cliques of weakly triangulated graphs. Keywords: treewidth, minimum ll-in, weakl...

We give a characterization of minimal triangulation of graphs using the notion of “maximal set of neighbor separators”. We prove that if all the maximal sets of neighbor separators of some graphs can be computed in polynomial time, the treewidth of those graphs can be computed in polynomial time. This notion also unifies the already known algorithm...

We prove that in an n-vertex graph, an induced planar subgraph of maximum size can be found in time O(1.7347 n ). This is the first algorithm breaking the trivial 2 n n O(1) bound of the brute-force search algorithm for the Maximum Induced Planar Subgraph problem.

Abstract We show that the treewidth and the minimum,fill-in of an n-vertex graph can be com- puted in time O(1.8899,). Keywords: Exact exponential algorithm, treewidth, fill-in, minimal separators, potential max-

Nous étudions les décompositions arborescentes de graphes et les paramètres de largeur associés (largeur arborescente, largeur linéaire) sous plusieurs aspects. Nous proposons des algorithmes pour le calcul de la largeur arborescente des graphes ayant une quantité polynomiale de séparateurs minimaux et des algorithmes exacts et d'approximation dans...