Anthony Perez

On the (Non-)existence of Polynomial Kernels for P l -free Edge Modification Problems

Given a graph G = (V , E) and an integer k, an edge modification problem for a graph property Π consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V , E △ F ) satisfies the property Π. In the Π edge-completion problem, the set F of edges is constrained to be disjoint from E; in the Π edge-deletion problem, F is a subset of E; no constraint is imposed on F in the Π edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F , It has been proved that (Cai, IPL:58(4)-1996) if Π is an hereditary property characterized by a finite set of forbidden induced subgraph, then the three Π edge-modification problems are fixed-parameter tractable. It was then natural to ask (Cai, IWPEC 2006) whether these Π edge-modification problems also admit a polynomial size kernel (i.e. any instance (G, k) can be reduced in polynomial time to an equivalent instance (G′ , k′ ) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlstr¨om (IWPEC 2009) answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraph. It is a challenging question to characterized for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlstr¨om asked whether the result holds when the forbidden subgraphs are paths and pointed out that the problem is already open in the case of P4 -free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for Pl -free graphs and Cl -free graph for large enough l.

Polynomial kernels for Proper Interval Completion and related problems

Given a graph G=(V,E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k pairs of (VxV)@?E such that the graph H=(V,E@?F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research. This problem is known to be FPT (Kaplan, Tarjan and Shamir, FOCS@?94), but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with O(k^3) vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem, admits a kernel with O(k^2) vertices, completing a previous result of Guo (ISAAC@?07).

Conflict packing yields linear vertex-kernels for k-FAST, k-dense RTI and a related problem

We develop a technique that we call Conflict Packing in the context of kernelization [7]. We illustrate this technique on several well-studied problems: FEEDBACK ARC SET IN TOURNAMENTS, DENSE ROOTED TRIPLET INCONSISTENCY and BETWEENNESS IN TOURNAMENTS. For the former, one is given a tournament T = (V, A) and seeks a set of at most k arcs whose reversal in T results in an acyclic tournament. While a linear vertex-kernel is already known for this problem [6], using the Conflict Packing allows us to find a so-called safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the DENSE ROOTED TRIPLET INCONSISTENCY problem, one is given a set of vertices V and a dense collection R of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from R. Using again the Conflict Packing, we prove that the DENSE ROOTED TRIPLET INCONSISTENCY problem admits a linear vertex-kernel. This result improves the best known bound of O(k2) vertices for this problem [16]. Finally, we use this technique to obtain a linear vertex-kernel for BETWEENNESS IN TOURNAMENTS, where one is given a set of vertices V and a dense collection R of betweenness triplets and seeks an ordering containing all but at most k triplets from R. To the best of our knowledge this result constitutes the first polynomial kernel for the problem.

Kernels for Feedback Arc Set In Tournaments

A tournament T = (V , A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed ϵ > 0, the kernelized instance has at most (2 + ϵ)k vertices. Our result improves the previous known bound of O(k2 ) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k- FAST.

On the importance of considering social capitalism when measuring influence on Twitter

Influence on Twitter has drawn a lot of attention these past few years since this microblogging service is used to share, seek or debate about any kind of information. Several tools providing so-called influential scores have thus been proposed. However, the algorithms behind them are kept secret and it is not clear how they consider influence. Yet, many users rely on such tools to evaluate and even try to improve their influence in the Twitter network. In a recent work, it has been shown that automatic accounts can obtain high influential scores with no intuitive reason. Extending and completing this work, we show that such measures fail at distinguishing so-called social capitalists from real, truthful users. This enlights the fact that actual scores do not seem to consider the way followers and interactions are obtained on the network. To overcome this issue, we define a classifier that discriminates social capitalists from truthful users. To that aim, we crawled the Twitter network to gather examples of certified social capitalists and regular users and obtained features related to the profile and behavior of each user. We then use such a classifier to balance Klouts score to adjust influential scores. We also developed an application that allows using our classifier online. We believe our work should raise the question of the legitimacy of influence on Twitter, and lead to significant improvements in the way it is measured.

Identifying the community roles of social capitalists in the Twitter network

In the context of Twitter, social capitalists are users trying to increase their number of followers and interactions by any means. They are not healthy for the service, because they introduce a bias in the way user influence and visibility are perceived. Understanding their behavior and position in the network is thus of important interest. In this work, we propose to do so by focusing on the community structure level. We first extend an existing method based on the notion of community role, on three different points: 1) handling of directed networks, 2) more precise modeling of the community-related connectivity and 3) unsupervised role identification. We then take advantage of an existing tool to detect social capitalists, and apply our method to analyze their organization and how their links spread across the network. The specific community roles they hold in the network let us know that they reach to obtain high visibility.

A reliable and evolutive web application to detect social capitalists

On Twitter, social capitalists use dedicated hashtags and mutual subscriptions to each other in order to gain followers and to be retweeted. Their methods are successful enough to make them appear as influent users. Indeed, applications dedicated to the influence measurement such as Klout and Kred give high scores to most of these users. Meanwhile, their high number of retweets and followers are not due to the relevance of the content they tweet, but to their social capitalism techniques. In order to be able to detect these users, we train a classifier using a dataset of social capitalists and regular users. We then implement this classifier in a web application that we call DDP. DDP allows users to test whether a Twitter account is a social capitalist or not and to visualize the data we use to make the prediction. DDP allows administrator to crawl data from a lot of users automatically. Furthermore, administrators can manually label Twitter accounts as social capitalists or regular users to add them into the dataset. Finally, administrators can train new classifiers in order to take into account the new Twitter accounts added to the dataset, and thus making evolve the classifier with these new recently collected data. The web application is thus a way to collect data, make evolve the knowledge about social capitalists and to keep detecting them efficiently.

Detecting Social Capitalists on Twitter Using Similarity Measures

Social networks such as Twitter or Facebook are part of the phenomenon called Big Data, a term used to describe very large and complex data sets. To represent these networks, the connections between users can be easily represented using (directed) graphs. In this paper, we are mainly focused on two different aspects of social network analysis. First, our goal is to find an efficient and high-level way to store and process a social network graph, using reasonable computing resources (processor and memory).We believe that this is an important research interest, since it provides a more democratic method to deal with large graphs.Next, we turn our attention to the study of social capitalists, a specific kind of users on Twitter. Roughly speaking, such users try to gain visibility by following other users regardless of their contents. Using two similarity measures called overlap index and ratio, we show that such users may be detected and classified very efficiently.

Exact Algorithms for Weak Roman Domination

We consider the Weak Roman Domination problem. Given an undirected graph G = (V,E), the aim is to find a weak roman domination function (wrd-function for short) of minimum cost, i.e. a function f: V → {0,1,2} such that every vertex v ∈ V is defended (i.e. there exists a neighbor u of v, possibly u = v, such that \(f(u) \geqslant 1\)) and for every vertex v ∈ V with f(v) = 0 there exists a neighbor u of v such that \(f(u) \geqslant 1\) and the function f u → v defined by:

Polynomial Kernels for 3-Leaf Power Graph Modification Problems

A graph G = (V,E) is a 3-leaf power iff there exists a tree T the leaf set of which is V and such that (u,v) ? E iff u and v are at distance at most 3 in T. The 3-leaf power edge modification problems, i.e. edition (also known as the Closest 3-Leaf Power), completion and edge-deletion are FPT when parameterized by the size of the edge set modification. However, a polynomial kernel was known for none of these three problems. For each of them, we provide a kernel with O(k 3) vertices that can be computed in linear time. We thereby answer an open question first mentioned by Dom, Guo, Huffner and Niedermeier [9].