164 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On linear, fractional, and submodular optimization
In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Data analysis with merge trees
Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram
Optimal distance query reconstruction for graphs without long induced cycles
Let be an -vertex connected graph of maximum degree .
Given access to and an oracle that given two vertices , returns
the shortest path distance between and , how many queries are needed to
reconstruct ? We give a simple deterministic algorithm to reconstruct trees
using distance queries and show that even
randomised algorithms need to use at least
queries in expectation. The best previous lower bound was an
information-theoretic lower bound of . Our lower
bound also extends to related query models including distance queries for
phylogenetic trees, membership queries for learning partitions and path queries
in directed trees.
We extend our deterministic algorithm to reconstruct graphs without induced
cycles of length at least using queries, which
includes various graph classes of interest such as chordal graphs, permutation
graphs and AT-free graphs. Since the previously best known randomised algorithm
for chordal graphs uses queries in expectation, we both
get rid off the randomness and get the optimal dependency in for chordal
graphs and various other graph classes.
Finally, we build on an algorithm of Kannan, Mathieu, and Zhou [ICALP, 2015]
to give a randomised algorithm for reconstructing graphs of treelength
using queries in expectation.Comment: 35 page
Simplicial bounded cohomology and stability
We introduce a set of combinatorial techniques for studying the simplicial
bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We
apply these methods to prove several new bounded acyclicity results for
semi-simplicial sets appearing in the homological stability literature. Our
strategy is to recast classical arguments (due to Bestvina, Maazen, van der
Kallen, Vogtmann, Charney and, recently, Galatius--Randal-Williams) in the
setting of bounded cohomology using uniformly bounded refinements of well-known
simplicial tools. Combined with ideas developed by Monod and De la Cruz
Mengual--Hartnick, we deduce slope- stability results for the bounded
cohomology of two large classes of linear groups: general linear groups over
any ring with finite Bass stable rank and certain automorphism groups of
quadratic modules over the integers or any field of characteristic zero. We
expect that many other results in the literature on homological stability admit
bounded cohomological analogues by applying the blueprint provided in this
work.Comment: 53 pages. Comments welcome
Combinatorics of the Permutahedra, Associahedra, and Friends
I present an overview of the research I have conducted for the past ten years
in algebraic, bijective, enumerative, and geometric combinatorics. The two main
objects I have studied are the permutahedron and the associahedron as well as
the two partial orders they are related to: the weak order on permutations and
the Tamari lattice. This document contains a general introduction (Chapters 1
and 2) on those objects which requires very little previous knowledge and
should be accessible to non-specialist such as master students. Chapters 3 to 8
present the research I have conducted and its general context. You will find:
* a presentation of the current knowledge on Tamari interval and a precise
description of the family of Tamari interval-posets which I have introduced
along with the rise-contact involution to prove the symmetry of the rises and
the contacts in Tamari intervals;
* my most recent results concerning q, t-enumeration of Catalan objects and
Tamari intervals in relation with triangular partitions;
* the descriptions of the integer poset lattice and integer poset Hopf
algebra and their relations to well known structures in algebraic
combinatorics;
* the construction of the permutree lattice, the permutree Hopf algebra and
permutreehedron;
* the construction of the s-weak order and s-permutahedron along with the
s-Tamari lattice and s-associahedron.
Chapter 9 is dedicated to the experimental method in combinatorics research
especially related to the SageMath software. Chapter 10 describes the outreach
efforts I have participated in and some of my approach towards mathematical
knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche
Parameterizing Path Partitions
We study the algorithmic complexity of partitioning the vertex set of a given
(di)graph into a small number of paths. The Path Partition problem (PP) has
been studied extensively, as it includes Hamiltonian Path as a special case.
The natural variants where the paths are required to be either \emph{induced}
(Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition,
SPP), have received much less attention. Both problems are known to be
NP-complete on undirected graphs; we strengthen this by showing that they
remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP
remains \NP-hard on undirected bipartite graphs. When parameterized by the
natural parameter ``number of paths'', both SPP and IPP are shown to be
W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected
graphs for the same parameter, as well as for other special subclasses of
directed graphs (IPP is known to be NP-hard on undirected graphs, even for two
paths). On the positive side, we show that for undirected graphs, both problems
are in FPT, parameterized by neighborhood diversity. We also give an explicit
algorithm for the vertex cover parameterization of PP. When considering the
dual parameterization (graph order minus number of paths), all three variants,
IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the
mentioned neighborhood diversity and dual parameterization results to directed
graphs; here, we need to define a proper novel notion of directed neighborhood
diversity. As we also show, most of our results also transfer to the case of
covering by edge-disjoint paths, and purely covering.Comment: 27 pages, 8 figures. A short version appeared in the proceedings of
the CIAC 2023 conferenc
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
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