of n and c a ﬁxed constant, could ever be devised.
In what follows, since there exist many natural parameters for a given problem (e.g., for a graph- problem, the value of an optimal solution, the size of a **minimum** **vertex** **cover** of the input-graph, the pathwidth or the treewidth of the input-graph, . . . , [14, 17]), we will use notation FPT(p) to denote that the ﬁxed parameter tractability considered is with respect to parameter p. In other words, the result by [8], just mentioned can be stated as max k-**vertex** **cover** / ∈ FPT(ℓ), unless FPT(ℓ) = W[1]. Let us note that, except for the parameters stated just above, max k-**vertex** **cover** admits another very natural parameter, the integer k.

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Related work In [10], it is shown that extension of partial solutions is NP-hard for computing prime implicants of the dual of a Boolean function; a problem which can also be seen as trying to find a minimal hitting set for the prime implicants of the input function. Interpreted in this way, the proof from [10] yields NP-hardness for the minimal extension problem for 3-Hitting Set. This result was extended in [6] to prove NP-hardness for the extension of minimal dominating sets (Ext DS), even restricted to planar cubic graphs.Similarly, it was shown in [5] that extension for **minimum** **vertex** **cover** restricted to planar cubic graphs is NP-hard. The first systematic study of this type of problems was exhibited in [11] providing quite a number of different examples of this type of problem.

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7. Asgeirsson, E., Stein, C.: **Vertex** **Cover** Approximation on Random Graphs. In: 6th Workshop on Experimental Algorithms. vol. LNCS 4525, pp. 285–296 (2007) 8. Bar-Yehuda, R., Hermelin, D., Rawitz, D.: **Minimum** **Vertex** **Cover** in Rectangle
Graphs. In: 18th Annual European Conference on Algorithms. pp. 255–266 (2010) 9. Bomze, I.M., Budinich, M., Pardalos, P.M., Pedillo, M.: Handbook of Combina- torial Optimization, chap. The Maximum Clique Problem, pp. 1–74. Kluwer Aca- demic Publishers (1999)

where n is the order of the graph G and α(G) its stability number (cardinality of a maximum independent set of G). We can suppose that the IS algorithm A ′′ , solving
approximately the instances of the family G, when applied to graphs not contained in G provides either solutions with ratio smaller than ρ (ρ ≤ 1) or non feasible solutions. Also we will denote by τ the cardinality of a **minimum** **vertex** **cover**.

To be able to give better preprocessing guarantees, one can use structural parameters which take on smaller values than the size of a **minimum** **vertex** **cover**, a quantity henceforth called the **vertex** **cover** number. Such structural parameterizations can conveniently be described in terms of the **vertex**-deletion distance to certain graph families F . Note that the **vertex** **cover** number vc(G) of G can be defined as the **minimum** number of **vertex** deletions needed to reduce G to an edgeless graph. Hence this number will always be at least as large as the feedback **vertex** number fvs(G) of G, which is the **vertex**-deletion distance of G to a forest. In 2011, it was shown that **Vertex** **Cover** even admits a polynomial kernelization when parameterized by the feedback **vertex** number [21, 22]. This triggered a long line of follow-up research, which aimed to find the most general graph families F such that **Vertex** **Cover** admits a polynomial kernelization when parameterized by **vertex**-deletion distance to F . Polynomial kernelizations were obtained for the families F of graphs of maximum degree two [27], of graphs of constant tree-depth [5, 23], of the pseudo-forests where each connected component has at most one cycle [17], and for d-quasi-forests in which each connected component has a feedback **vertex** set of size at most d ∈ O(1) [18]. Note that all these target graph classes are closed under taking minors. Using randomized

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Quick Overview of Existing Algorithms for **Vertex** Covering. Many algorithms have been proposed for the **vertex** **cover** problem. As it is NP-hard, most of the methods are approximation algorithms or heuristics. Here, we give a rapid overview of these methods.
A well-known heuristic is to select a **vertex** of maximum degree and delete this **vertex** and its incident edges from the graph, until all edges have been removed [10]. It has an approximation ratio in O(log ∆). Another popular algorithm, with the best known constant approximation ratio, 2, is to construct a maximal matching of the input graph and return the vertices of the matching (see [3]). To compute such a solution, an edge is randomly chosen, and its two endpoints with their incident edges are deleted from the graph, until all edges have been removed. For these algorithms, in order to delete a **vertex** and its incident edges, we have to modify the input graph or store information on deleted elements into the memory of the processing unit, and that does not satisfy constraints C 1 and C 2 . Another well-known algorithm is to construct a DFS spanning tree and

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2 . Obviously, if H consists of a single triangle, then ψ3(H) = 1 ≤ 3 2 = n
2 . Assume H has at least four vertices. Since H is a 2-connected outerpla- nar graph, the closed trail bounding the outer face contains each **vertex** of H precisely once, hence H is Hamiltonian. Let v1, v2, . . . , vn be the cyclic ordering of vertices of H along the Hamiltonian cycle. Colour a **vertex** vi white, if the degree of vi in H is 2, otherwise colour it black. Since all the inner faces of H are triangles, there are no two consecutive white vertices, unless H consists of a single triangle, which is excluded. Hence, the white vertices induce an independent set. The edge vivi+1 is called good, if it has a white endvertex, otherwise it is bad. If all the edges incident with the outer faces are good, it implies that n is even, and half of the vertices are white. Then the set of all black vertices is a 3-path **vertex** **cover** of size n

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The New Approach. In this paper, we provide a completely new approach for analyzing the remaining **vertex** connectiv- ity after **vertex** sampling. Instead of sampling with proba- bility 1/ √ k in each phase, we sample with a lower proba- bility of 1/k. In addition, rather than directly showing that the sampled nodes induce a connected subgraph, we intro- duce the notion of λ-semi-connectivity which allows us to analyze the progress in a more refined way. We call a ver- tex set S ⊆ V λ-semi-connected if for every partition of the connected components of the induced subgraph G[S] into two parts, there are λ nodes in V \ S which are adjacent to components on both sides of the partition. Roughly, in each phase, each component of G[S] in expectation gains at least one new such connector node to another component, and we use this to show that within t+log n phases, the semi- connectivity of the sampled set of nodes grows by Ω(t). It follows that when sampling with probability p, we get semi- connectivity at least Ω(kp). This is the main technical con- tribution of our paper, and it is shown by carefully analyz- ing how the semi-connectivity of the sampled set grows by adding new random vertices. Having now a set with semi- connectivity Ω(kp), we can use techniques similar to the ones used in [1, 6] to show that another round of sampling with probability p yields a connected graph with probability 1 − e −Θ(kp 2 ) (as long as kp 2 = Ω(log n)).

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Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024.. On-line vertex-covering Marc Demange, Vangelis Th.[r]

problem, we set the time limit of either IT1 TL or IT2 TL, where IT2 TL is used only if the total time spent is greater than TOTAL TL. In the case that no optimal solution is found, we leave the corresponding column temporarily un- covered and use the lower bound obtained by the unterminated MIP as a lower bound on the **minimum** local ratio of the **cover**. If after the last iteration there are uncovered columns, we **cover** these columns with the best ERs that **cover** them and report, in column NO, how many columns were covered in this way.

Front **Cover**: “My Maximum Security Home”
2013, white paper with black pencil sketch Jeannette Tossounian
“Apparently there’s a huge snow storm today. Of course I can’t see out my frosted windows, but there’s the shadow of snow on the ledge over a foot high. All programs were cancelled today and there was no mail.

T (n − 1) + T (n − 11) + p(n). This leads to T (n) = O ∗ (1.185 n ).
Note that the above algorithm can be generalized to find a 1/k-approximation algorithm (for any integer k) in time T (n) 6 T (n −1)+T (n−7k+3)+p(n). Obviously, improved running times would follow from considering, for example, either modifications of algorithms more sophisticated than the one presented in this section, or a more efficient counting technique such as the one presented in [17]. However, up to now, the techniques presented in next sections give better results. This is not the case, for instance, for min set **cover**, where approximate pruning of the search tree ([7, 9]) achieves very interesting results.

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there are 2 s−1 −1 nonempty subsets of A, and s−1 = ⌈log 2 (k +1)⌉. Without loss of generality, we can choose the sets A i in such a way that the graph
constructed so far is connected.
Clearly the auxiliary vertices form a 1-identifying code in this graph: the 1-identifying set of each auxiliary **vertex** is a singleton consisting of the **vertex** itself; and for all the vertices x i , the 1-identifying set contains a 1

111 **COVER** ART
Tim Felfoldi is currently a prisoner at Collins Bay Institution (**minimum**) and is working on a book that features his art work, including the pieces on the front and back covers of this issue of the Journal of Prisoners on Prisons. Below is his biography:

Front **Cover**: “Spider & Fly” – August 10th Illustration (2004)
Peter Collins
Since August 10, 1975, Prison Justice Day (PJD) has been observed annually in Canada. The movement began in Millhaven Institution to commemorate “the first anniversary of the death of Eddie Nalon, who had committed suicide while in solitary confinement in Millhaven’s SHU [Special Handling Unit]” (Gaucher, 1991, JPP Volume 3, p. 98). Over the years, PJD has been instrumental in promoting the human rights of prisoners including the right to freedom of speech (see www.prisonjustice.ca).

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Although the goal of our empirical study was not to show a clear superior- ity in terms of clustering but rather to assess the **vertex**-**vertex** dissimilarities obtained by GT, we showed that our proposed approach is competitive with well- known clustering methods, Louvain and MCL. We also showed that by comput- ing forests of graph trees and other trees that specialize in other types of input data, e.g, feature vectors, it is then possible to compute pairwise dissimilarities between vertices in attributed graphs.

we obtained using our approach are not consistently better in this first assess- ment, the methods still seems to give similar results without any fine tuning.
5 Discussion and future work
In this paper, we presented a method based on the construction of random trees to compute dissimilarities between graph vertices, called GT. For **vertex** clustering purposes, our proposed approach is plug-and-play, since any clustering algorithm that can work on a dissimilarity matrix can then be used. Moreover, it could find application beyond graphs, for instance in relational structures in general.

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WTZARD, which produces accurate lire drawings of ccomlex scenes <Visior Flash forthcoring >. Wizard is capable of handling partial informatior when rroducing lire dr[r]

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]