## Séminaires du Pôle 2 : "Optimisation combinatoire, algorithmique"

publié le

### Liste de diffusion.

Possibilité de s’inscrire à la liste de diffusion en envoyant un email à :
jerome. monnot @ dauphine . fr

Les séminaires ont lieu en général le lundi à 14h.

### lundi 9 avril 2018, 14h, : Vadim V. Lozin (University of Warwick) : From structure to algorithms : breaking the boundaries

Many algorithmic problems that are generally intractable may become
easy when restricted to instances of particular structure. When and why
a difficult problem becomes easy ? To answer these questions, we employ
the notion of boundary classes of graphs. In this talk, we focus on the maximum
independent set problem and shed some light on the structure of the boundary
separating difficult instances of this problem from polynomially solvable ones.
We also analyze algorithmic tools to break the boundary and discuss
similar questions with respect to some other algorithmic problems, including
Satisfiability, the central problem of Theoretical Computer Science.

### lundi 26 mars 2018, 14h, P301 : Magnus Wahlström (Royal Holloway, London) : FPT-algorithms via LP-relaxations

LP-relaxations are traditionally (within theoretical computer science)
used for computing approximate solutions to NP-hard problems, but
across the last few years there have been several examples where
LP-relaxations have been used to guide FPT-algorithms — that is,
exact (non-approximate) algorithms whose running time is bounded in
terms of a "parameter" of the input instance. This approach has given
algorithms that are simultaneously simpler and faster for a range of
central problems in parameterized complexity. At the same time, this
is applicable only to specific problems and relaxations ; an arbitrary
LP-relaxation, even one that has good properties in an approximation
sense, will in general give no useful guidance with respect to exact
solutions.

In this talk, we give an overview of these FPT applications, and the
conditions they impose on the LP-relaxation. Our main focus is on an
approach of "discrete relaxation" referred to as Valued CSPs, but we
also briefly survey more immediately combinatorial conditions, as well
as related algorithms that solve these problems more efficiently
(e.g., so-called linear-time FPT algorithms) by bypassing the
LP-solver.

### lundi 26 fevrier 2018, 14h, A407 : Henning Fernau (Theoretical Computer Science in Trier) : Self-monitoring approximation algorithms

Reduction rules are one of the key techniques for the design of parameterized algorithms.
They can be seen as formalizing a certain kind of heuristic approach to solving hard combinatorial problems.
We propose to use such a strategy in the area of approximation algorithms.
One of the features that we may gain is a self-monitoring property.
This means that the algorithm that is run on a given instance $I$ can predict an approximation factor of the solution produced on $I$ that
is often (according to experiments) far better than the theoretical estimate that is based on a worst-case analysis.

Bibliography :

[1] F. N. Abu-Khzam, C. Bazgan, M. Chopin, and H. Fernau.
Data reductions and combinatorial bounds for improved approximation
algorithms. Journal of Computer and System Sciences, 82:503—520, 2016.

[2] L. Brankovic and H. Fernau.
A novel parameterised approximation algorithm for minimum vertex cover.
Theoretical Computer Science, 511:85—108, 2013.

### lundi 11 décembre 2017, 14h, A 304 : Ararat Harutyunyan (LAMSADE) : A disproof of the normal graphs conjecture

Abstract : A graph G is called normal if there exist two coverings, C and S, of its vertex set such that every member of C induces a clique in G, every member of S induces an independent set in G, and any clique in C and independent set in S have a non-empty intersection. Normal graphs derive their motivation from information theory, where they are related to the Shannon capacity of a graph. In particular, they form a family which extends the class of perfect graphs. It was conjectured by De Simone and Körner [DAM ’99] that a graph G is normal if G does not contain C_5, C_7 and the complement of C_7 as an induced subgraph. Using random graphs and rather routine probabilistic methods, we give a disproof of this conjecture.

Joint work with Lucas Pastor (G-SCOP, Grenoble) and Stéphan Thomassé (ENS Lyon).

### lundi 20 novembre 2017, 14h, P 301 : Julien Baste (LIP6) : F-M-DELETION parameterized by treewidth

For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a set S of vertices of G of size at most k such that G without the vertices of S does not contain any of the graphs of F as a minor. This problem is a generalization of some well known problems as VERTEX COVER (F=K_2), FEEDBACK VERTEX SET (F=K_3), or VERTEX PLANARIZATION (F=K_5, K_3,3 ). We are interested in the parameterized complexity of F-M-DELETION when the parameter is the treewidth of the input graph, denoted by tw. Our objective is to determine, for a fixed F, the smallest function f such that F-M-DELETION can be solved in time f(tw)*poly(n) on n-vertex graphs.

### lundi 13 novembre 2017, 14h, salle A (C206, 2ème étage) : Tomáš Toufar (Charles University, Czech Republic) : Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Slides

We study the Steiner Tree problem, in which a set of terminal vertices
needs to be connected in the cheapest possible way in an edge-weighted
graph. This problem has been extensively studied from the viewpoint of
approximation and also parametrization. In particular, on one hand
Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if
parameterized by the number of non-terminals (Steiner vertices) in the
optimum solution. In contrast to this we
give an efficient parameterized approximation scheme (EPAS), which
circumvents both hardness results. Moreover, our methods imply the
existence of a polynomial size approximate kernelization scheme
(PSAKS) for the assumed parameter.
We further study the parameterized approximability of other variants
of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For
neither of these an EPAS is likely to exist for the studied parameter :
for Steiner Forest an easy observation shows that the problem is
APX-hard, even if the input graph contains no Steiner vertices. For
Directed Steiner Tree we prove that computing a constant approximation
for this parameter is W[1]-hard. Nevertheless, we
show that an EPAS exists for Unweighted Directed Steiner Tree. Also we
prove that there is an EPAS and a PSAKS for Steiner Forest if in
addition to the number of Steiner vertices, the number of connected
components of an optimal solution is considered to be a parameter.

### lundi 16 octobre 2017, 14h, salle A (2ème étage) : Eunjung Kim (LAMSADE) : Erdos-Posa Property of Chordless Cycles and its Applications

A chordless cycle is a cycle of length at least 4 that has no chord. We prove that the class of all chordless cycles has the Erdos-Posa property, which resolves the major open question concerning the Erdos-Posa property. We complement our main result by showing that the class of all chordless cycles of length at least l for any fixed l ≥ 5 does not have the Erdos-Posa property.

Our proof for chordless cycles is constructive : in polynomial time, one can either find k + 1 vertex-disjoint chordless cycles, or ck2 log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(opt log opt) for Chordal Vertex Deletion, which improves the best known approximation by Agrawal et. al. The improved approximation algorithm entails improvement over the known kernelization for Chordal Vertex Deletion.

As a corollary, for a non-negative integral function w defined on the vertex set of a graph G, the minimum value \sum_x\in S w(x) over all vertex sets S where G − S is forest is at most O(k2 log k) where k is the maximum number of cycles (not necessarily vertex-disjoint) in G such that each vertex v is used at most w(v) times.

### lundi 2 octobre 2017, 14h, P303 : Giuseppe F. Italiano (Universita’ di Roma "Tor Vergata") : 2-Connectivity in Directed Graphs

We survey some recent results on 2-edge and 2-vertex
connectivity in directed graphs. Despite being complete analogs of the
corresponding notions on undirected graphs, in digraphs 2-connectivity
has a much richer and more complicated structure. For undirected
graphs it has been known for over 40 years how to compute all bridges,
articulation points, 2-edge- and 2-vertex-connected components in
linear time, by simply using depth first search. In the case of
digraphs, however, the very same problems have been much more
challenging and have been tackled only very recently.

Pour les exposés antérieurs, voir cette page.