Date : March 29, 2023, 1:30 p.m. - Room :Salle A211

Introduction to tree-independence number

Clément DALLARD, ATER - LIFO (Orléans)

Tree decompositions are a common and useful data structure for designing dynamic programming algorithms for graph problems. In order to obtain efficient algorithms, one looks for a tree decomposition of the input graph that minimizes some measure on the subgraphs induced by the bags. For instance, when this measure is the number of vertices, we obtain the well-known notion of treewidth. In this talk, the considered measure is the independence number.
The independence number of a tree decomposition of a graph G is the smallest integer k such that each bag induces a subgraph with independence number at most k. The tree-independence number of G is defined as the minimum independence number over all tree decompositions of G. Given a tree decomposition with bounded independence number, the Maximum Weight Independent Set and several related NP-hard problems can be solved in polynomial time.

We will discuss how to compute tree decompositions with bounded independence number efficiently (when they exist), the algorithmic use of such tree decompositions, and the strong relationship with the notion of (tw,ω)-boundedness.

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