Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on Theoretical Computer Science (ICTCS'2007

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}. For a class F of boolean functions and a class G of graphs, an (F, G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #Pcomplete to compute the number of fixed points in an (F, G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas

Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications

Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable

Η Πολυωνυμική Ιεραρχία Συμβουλής και το Θεώρημα του Dinneen

H παρούσα διπλωματική εργασία ξεκίνησε και επεκτάθηκε με αφορμή και γνώμονα ένα θεώρημα του Dinneen, το οποίο αποτελεί μέρος της μελέτης του ρυθμού αυξησης του πληθάριθμου του συνόλου παρεμποδίσεων οικογενειών γραφημάτων, κλειστών ως προς ελάσσονα, τα οποία βασίζονται σε παραμετρικά προβλήματα. Επιπλέον συνδυάζει διαφο- ρετικούς τομείς των μαθηματικών, μιας και συσχετίζει την πολυπλοκότητα των προαναφερθέντων προβλημάτων, με την Πολυωνυμική Ιεραρχία και την κατάρρευση αυτής. Ο βασικός μας στόχος είναι η σύνοψη του συνόλου των μαθηματικών εννοιών και κεφαλαίων των μαθηματικών, πάνω στα οποία βασίζεται το θεώρημα αλλά και η περαιτέρω παρουσίαση και ανάλυση κάποιων θεωρημάτων και προτάσεων, που σχετίζονται άμεσα με αυτό, και αποτελούν εξεζητημένα θέματα της Υπολογιστικής Πολυπλοκότητας. Τα τελευταία, κυρίως, συνδέουν την κατάρρευση της Πολυωνυμικής Ιεραρχίας Συμβουλής με την κατάρρεση της Πολυωνυμικής Ιεραρχίας. Για να γίνουν σαφή όμως τα παραπάνω, απαιτείται η γνώση των Πολυωνυμικών Ιεραρχιών, της Kυκλωματικής Πολυπλοκότητας και πώς αυτή σχετίζεται με την κλασική Πολυπλοκότητα. Ο θεμέλιος λίθος και το βασικό εργαλείο για να αναπτυχθούν οι παραπάνω έννοιες αποτελούν οι Μηχανές Turing, που παρουσιάζονται στην αρχή. Ουσιαστικά δηλαδή η συλλογιστική πορεία της παρουσίασης είναι αντίστροφη του σχεδιασμού. Αυτό γίνεται χάρην ευκολίας της ανάγνωσης και καταγραφής, αλλά και για να παρουσιαστούν οι έννοιες με την φυσική τους σειρά.The present dissertation began and expanded on the occasion and in the light by one of Dinneen's theorems, which, is part of the study of the growth rate of the number obstructions for families of graphs, which closed under minors, are based on parameterized problems. Furthermore, combines different reticular files of mathematics since it correlates the complexity of the above alleged problems, with the Polynomial Hierarchy and its collapse. Our main goal is to summarize all the mathematical concepts and chapters of mathematics on which the theorem is based, as well as the further presentation and analysis of some theorems and propositions that are directly related to it and are sophisticated topics of Computational Complexity. The latter, in patricular, links the collapse of the Polynomial Hierarchy of Advice with the collapse of the Polynomial Hierarchy. However, in order to make the above clear, knowledge of Polynomial Hierarchies, Circuit Complexity and how it relates to classical Complexity is required. The foundatiton stone and the basic tool for developing the above concepts are the Turing Machines, presented at the beginning. Essentially the reasoning course of the presentation is the inverse of the design. This is done for the sake of ease of reading and recording dut in order to also present the concepts in their natural order

Theoretical and algorithmic approaches to field-programmable gate array partitioning

Many practical problems dealing with the design of Very Large Scale Integrated (VLSI) circuits can be modeled as graphs in which vertices represent components of the circuit and edges represent a relationship between these components. When expressed as graphs, these problems can then often be solved using graph theoretic methods. Unfortunately, many such problems are NP-complete, hence no practical exact solutions are known to exist. In this dissertation, we study NP-complete problems taken from the realm of partitioning for Field-Programmable Gate Arrays (FPGAs). We adopt a two-pronged approach of theory and practice, developing practical heuristics driven by theoretical study. The theoretical approach is motivated by well-quasi-order (WQO) theory, which can be used to show, among other things, that when some hard problems have fixed parameters, polynomial-time solutions exist. This is of significance in the area of FPGA partitioning, in which practical problems are often characterized by fixed parameter instances. WQO techniques are not generally practical, however, and we develop new methods to solve several important problems in VLSI that are not even amenable to WQO techniques. We begin with a representative partitioning problem, Min Degree Graph Partition (MDGP), the fixed-parameter version of which is closed under the immersion order. \Ve show that the obstruction set ( set of immersion minimal elements) for this problem is computable; we prove both upper and lower bounds on the obstruction set size; and we completely characterize all fixed-parameter MDGP simple tree obstructions. WQO theory tells us only that fixed-parameter MDGP is solvable in (high-degree) polynomial time. We attack the problem using what we refer to as kd-candidate subsets, culminating in linear-time decision and search algorithms. The kd-candidate subset method also paves the way for an efficient heuristic for the FPGA Minimization problem. We then broaden our scope to incorporate delay minimization into FPGA partitioning. We develop, analyze and test a novel method called critical path compression, inspired in part by compiler optimization techniques. We then look at a variety of generalizations of MDGP. Some of these problems are not immersion closed; others are not even defined in a way that WQO theory applies. However, almost all of them are efficiently solvable via the kd-candidate subset approach. Interspersed in these results are many refinements of what is known about the complexity of these problems. We also discuss a few other solution strategies, and present many open problems