12 research outputs found
A Linear Time Algorithm for Counting #2SAT on Series-Parallel Formulas
CapĂtulo de libroAn O(m + n) time algorithm is presented for counting the
number of models of a two Conjunctive Normal Form Boolean Formula
whose constrained graph is represented by a Series-Parallel graph, where
n is the number of variables and m is the number of clauses. To the
best of our knowledge, no linear time algorithm has been developed for
counting in this kind of formulas
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces
This thesis comprises three apparently very independent parts. However, there is a unity behind I would like to sketch very briefly.
Formally graphs are in the background of most chapters and so is the duality local versus global. The first section is concerned with globally coloring graphs under some local assumptions. Algorithmically it is an intrinsically difficult task and neural networks, the topic of the second part can be used to approach intractable problems. Simple local interactions with emergent collective behavior are one of the essential features of these networks. Their current models are similar to some of those encountered in statistical mechanics, like spin glasses. In the third part, we study ultrametricity, a concept recently rediscovered by theoretical physicists in the analysis of spin-glasses. Ultrametricity can be expressed as a local constraint on the shape of each triangle of the given metric space.
Unless otherwise stated, results in the first and second part are essentially original. Since the third part represents a joint work with Michael Aschbacher, Eric Baum and Richard Wilson, I should perhaps try to outline my contribution though paternity of collective results is somewhat fuzzy. While working on neural networks and spin glasses Eric and I got interested in ultrametricity. Several of us had found an initial polynomial upper bound, but the final results of "n + 1" was first reached independently by Michael and Richard. I think I obtained the theorems: 4.5, 6.1, 6.3 (using an idea of Eric), 6.4, 6.5, 6.6, 6.7 (with Richard and helpful references from Bruce Rothschild and Olga Taussky) and participated in some other results.</p
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of eďŹcient solutions and algorithms for computationally diďŹcult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
Schematics of Graphs and Hypergraphs
Graphenzeichnen als ein Teilgebiet der Informatik befasst sich mit dem Ziel Graphen oder deren Verallgemeinerung Hypergraphen geometrisch zu realisieren. Beschränkt man sich dabei auf visuelles Hervorheben von wesentlichen Informationen in Zeichenmodellen, spricht man von Schemata. Hauptinstrumente sind Konstruktionsalgorithmen und Charakterisierungen von Graphenklassen, die fĂźr die Konstruktion geeignet sind. In dieser Arbeit werden Schemata fĂźr Graphen und Hypergraphen formalisiert und mit den genannten Instrumenten untersucht. In der Dissertation wird zunächst das âpartial edge drawingâ (kurz: PED) Modell fĂźr Graphen (bezĂźglich gradliniger Zeichnung) untersucht. Dabei wird um Kreuzungen im Zentrum der Kante visuell zu eliminieren jede Kante durch ein kreuzungsfreies TeilstĂźck (= Stummel) am Start- und am Zielknoten ersetzt. Als Standard hat sich eine PED-Variante etabliert, in der das Längenverhältnis zwischen Stummel und Kante genau 1â4 ist (kurz: 1â4-SHPED). FĂźr 1â4-SHPEDs werden Konstruktionsalgorithmen, Klassifizierung, Implementierung und Evaluation präsentiert. AuĂerdem werden PED-Varianten mit festen Knotenpositionen und auf Basis orthogonaler Zeichnungen erforscht. Danach wird das BUS Modell fĂźr Hypergraphen untersucht, in welchem Hyperkanten durch fette horizontale oder vertikale â als BUS bezeichnete â Segmente repräsentiert werden. Dazu wird eine vollständige Charakterisierung von planaren Inzidenzgraphen von Hypergraphen angegeben, die eine planare Zeichnung im BUS Modell besitzen, und diverse planare BUS-Varianten mit festen Knotenpositionen werden diskutiert. Zum Schluss wird erstmals eine Punktmenge von subquadratischer GrĂśĂe angegeben, die eine planare Einbettung (Knoten werden auf Punkte abgebildet) von 2-auĂenplanaren Graphen ermĂśglicht
Reconfiguring Triangulations
The results in this thesis lie at the confluence of triangulations and reconfiguration. We make the observation that certain solved and unsolved problems about triangulations can be cast as reconfiguration problems. We then solve some reconfiguration problems that provide us new insights about triangulations. Following are the main contributions of this thesis:
1. We show that computing the flip distance between two triangulations of a point set is NP-complete. A flip is an operation that changes one triangulation into another by replacing one diagonal of a convex quadrilateral by the other diagonal. The flip distance, then, is the smallest number of flips needed to transform one triangulation into another. For the special case when the points are in convex position, the problem of computing the flip distance is a long-standing open problem.
2. Inspired by the problem of computing the flip distance, we start an investigation into computing shortest reconfiguration paths in reconfiguration graphs. We consider the reconfiguration graph of satisfying assignments of Boolean formulas where there is a node for each satisfying assignment of a formula and an edge whenever one assignment can be changed to another by changing the value of exactly one variable from 0 to 1 or from 1 to 0. We show that computing the shortest path between two satisfying assignments in the reconfiguration graph is either in P, NP-complete, or PSPACE-complete depending on the class the Boolean formula lies in.
3. We initiate the study of labelled reconfiguration. For the case of triangulations, we assign a unique label to each edge of the triangulation and a flip of an edge from e to e' assigns the same label to e' as e. We show that adding labels may make the reconfiguration graph disconnected. We also show that the worst-case reconfiguration distance changes when we assign labels. We show tight bounds on the worst case reconfiguration distance for edge-labelled triangulations of a convex polygon and of a spiral polygon, and edge-labelled spanning trees of a graph. We generalize the result on spanning trees to labelled bases of a matroid and show non-trivial upper bounds on the reconfiguration distance
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum