4 research outputs found
A tight Monte-Carlo algorithm for Steiner Tree parameterized by clique-width
Recently, Hegerfeld and Kratsch [ESA 2023] obtained the first tight
algorithmic results for hard connectivity problems parameterized by
clique-width. Concretely, they gave one-sided error Monte-Carlo algorithms that
given a -clique-expression solve Connected Vertex Cover in time
and Connected Dominating Set in time . Moreover,
under the Strong Exponential-Time Hypothesis (SETH) these results were showed
to be tight. However, they leave open several important benchmark problems,
whose complexity relative to treewidth had been settled by Cygan et al. [SODA
2011 & TALG 2018]. Among which is the Steiner Tree problem. As a key
obstruction they point out the exponential gap between the rank of certain
compatibility matrices, which is often used for algorithms, and the largest
triangular submatrix therein, which is essential for current lower bound
methods. Concretely, for Steiner Tree the -rank is , while no
triangular submatrix larger than was known. This yields time
, while the obtainable impossibility of time
under SETH was already known relative to pathwidth.
We close this gap by showing that Steiner Tree can be solved in time
given a -clique-expression. Hence, for all parameters between
cutwidth and clique-width it has the same tight complexity. We first show that
there is a ``representative submatrix'' of GF(2)-rank (ruling out larger
triangular submatrices). At first glance, this only allows to count (modulo 2)
the number of representations of valid solutions, but not the number of
solutions (even if a unique solution exists). We show how to overcome this
problem by isolating a unique representative of a unique solution, if one
exists. We believe that our approach will be instrumental for settling further
open problems in this research program
The bidimensionality theory and its algorithmic applications
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 201-219).Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(kÂČ), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results.(cont.) The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into Lâ (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O[sq. root( log n)] approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be [theta][sq. root(log n)]. We also prove various approximate max-flow/min-vertex- cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O[sq. root (log n)] pseudo-approximation for finding balanced vertex separators in general graphs.(cont.) Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k[sq. root(log k)]), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.by MohammadTaghi Hajiaghayi.Ph.D
Proceedings of the European Conference on Agricultural Engineering AgEng2021
This proceedings book results from the AgEng2021 Agricultural Engineering Conference under auspices of the European Society of Agricultural Engineers, held in an online format based on the University of Ăvora,
Portugal, from 4 to 8 July 2021.
This book contains the full papers of a selection of abstracts that were the base for the oral presentations and posters presented at the conference.
Presentations were distributed in eleven thematic areas: Artificial Intelligence, data processing and
management; Automation, robotics and sensor technology; Circular Economy; Education and Rural development; Energy and bioenergy; Integrated and sustainable Farming systems; New application
technologies and mechanisation; Post-harvest technologies; Smart farming / Precision agriculture; Soil, land and water engineering; Sustainable production in Farm buildings