79 research outputs found
Some relations of subsequences in permutations to graph theory with algorithmic applications
A thesis submitted to the Faculty dÂŁ Science of the University
of the Witwatersfand in fulfillment of the requirements for the
degree of Doctor of Philosophy.
Johannesburg, 1977.The representation of some types of graphs as permutations,
is utilized in devising efficient algorithms on those graphs.
Maximum 'cliques in permutation graphs and circle graphs
are found, by searching for a longest ascending or descending
subsequence in their representing permutation.
The correspondence between n-noded binary trees and
the set SSn of stack-sortable permutations, forms the
basis of an algorithm for generating and indexing such trees.
The-relations between a graph and its representing
p ermutation, are also employed in the proof of theorems
concerning properties of subsequences in this permutation.
In particular, expressions for the average lengths of the
longest ascending and descending subsequence a in a random
member of SSn , and the average number of inversions in such
a permutation, are derived using properties of binary trees.
Finally, a correspondence between the set SSn , and the set
of permutations of order n With no descending subsequence of
length 3, is demonstrated
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Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their
properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic
and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization,
Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions.
This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
Optimales Sortieren von Objekten
This thesis is concerned with the problem of optimally rearranging objects, in particular, railcars in a rail yard. The work is motivated by a research project of the Institute of Mathematical Optimization at Technische UniversitĂ€t Braunschweig, together with our project partner BASF, The Chemical Company, in Ludwigshafen. For many variants of such rearrangement problems - including the real-world application at BASF - we state the computational complexity by exploiting their equivalence to particular graph coloring, scheduling, and bin packing problems. We present mathematical optimization methods for determining schedules that are either optimal or close to optimal, and computational results are discussed from both a theoretical and practical point of view. In addition to the railway industry, there are other fields of application in which efficiently rearranging, sorting, or stacking is an important issue. For instance, the results obtained in this thesis could also be applied to solving certain piling problems in warehouses or container terminals.Die Dissertation beschĂ€ftigt sich mit dem optimalen Sortieren von Objekten, insbesondere von GĂŒterwagen in Rangierbahnhöfen. Motiviert wurde diese Arbeit durch ein BMBF-gefördertes Projekt mit der BASF, The Chemical Company, in Ludwigshafen. Zahlreiche Varianten derartiger Sortierprobleme werden mathematisch formuliert und komplexitĂ€tstheoretisch eingeordnet. FĂŒr viele Varianten wird deren Ăquivalenz zu bestimmten GraphenfĂ€rbungs-, Scheduling- sowie Bin-Packing-Problemen gezeigt. FĂŒr mehrere als theoretisch schwer bewiesene FĂ€lle werden schnelle approximative Algorithmen vorgeschlagen, die Lösungen mit einer beweisbaren GĂŒte liefern. Neben heuristischen Methoden werden auch exakte Verfahren zur Bestimmung optimaler Lösungen vorgestellt. Unter anderem handelt es sich bei den eingesetzten exakten AnsĂ€tzen um LP- sowie Lagrange-basierte Branch-and-Bound-Verfahren, die auf verschiedenen binĂ€ren Modellen beruhen. Die Lösungsmethoden werden durch die Auswertung von Rechenergebnissen fĂŒr reale Daten evaluiert. Den Abschluss der Dissertation bildet eine KompetitivitĂ€tsanalyse diverser Online-Varianten, die dadurch gekennzeichnet sind, dass nicht alle relevanten Informationen zu Beginn der Planung vorliegen. Es sei auf das Verwertungspotenzial der in dieser Arbeit vorgestellten Optimierungsverfahren innerhalb anderer Anwendungsbereiche, in denen Sortieren, Stapeln, Lagern oder Verstauen eine Rolle spielen, hingewiesen
NP-Completeness Results for Graph Burning on Geometric Graphs
Graph burning runs on discrete time steps. The aim is to burn all the
vertices in a given graph in the least number of time steps. This number is
known to be the burning number of the graph. The spread of social influence, an
alarm, or a social contagion can be modeled using graph burning. The less the
burning number, the faster the spread.
Optimal burning of general graphs is NP-Hard. There is a 3-approximation
algorithm to burn general graphs where as better approximation factors are
there for many sub classes. Here we study burning of grids; provide a lower
bound for burning arbitrary grids and a 2-approximation algorithm for burning
square grids. On the other hand, burning path forests, spider graphs, and trees
with maximum degree three is already known to be NP-Complete. In this article
we show burning problem to be NP-Complete on connected interval graphs,
permutation graphs and several other geometric graph classes as corollaries.Comment: 17 pages, 5 figure
Mod-phi convergence I: Normality zones and precise deviations
In this paper, we use the framework of mod- convergence to prove
precise large or moderate deviations for quite general sequences of real valued
random variables , which can be lattice or
non-lattice distributed. We establish precise estimates of the fluctuations
, instead of the usual estimates for the rate of
exponential decay . Our approach provides us with a
systematic way to characterise the normality zone, that is the zone in which
the Gaussian approximation for the tails is still valid. Besides, the residue
function measures the extent to which this approximation fails to hold at the
edge of the normality zone.
The first sections of the article are devoted to a proof of these abstract
results and comparisons with existing results. We then propose new examples
covered by this theory and coming from various areas of mathematics: classical
probability theory, number theory (statistics of additive arithmetic
functions), combinatorics (statistics of random permutations), random matrix
theory (characteristic polynomials of random matrices in compact Lie groups),
graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and
non-commutative probability theory (asymptotics of random character values of
symmetric groups). In particular, we complete our theory of precise deviations
by a concrete method of cumulants and dependency graphs, which applies to many
examples of sums of "weakly dependent" random variables. The large number as
well as the variety of examples hint at a universality class for second order
fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new
section on mod-Gaussian convergence coming from the factorization of the
generating function ; the multi-dimensional results have been moved to a
forthcoming paper ; and the introduction has been reworke
Multivariate Fine-Grained Complexity of Longest Common Subsequence
We revisit the classic combinatorial pattern matching problem of finding a
longest common subsequence (LCS). For strings and of length , a
textbook algorithm solves LCS in time , but although much effort has
been spent, no -time algorithm is known. Recent work
indeed shows that such an algorithm would refute the Strong Exponential Time
Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann,
K\"unnemann FOCS'15].
Despite the quadratic-time barrier, for over 40 years an enduring scientific
interest continued to produce fast algorithms for LCS and its variations.
Particular attention was put into identifying and exploiting input parameters
that yield strongly subquadratic time algorithms for special cases of interest,
e.g., differential file comparison. This line of research was successfully
pursued until 1990, at which time significant improvements came to a halt. In
this paper, using the lens of fine-grained complexity, our goal is to (1)
justify the lack of further improvements and (2) determine whether some special
cases of LCS admit faster algorithms than currently known.
To this end, we provide a systematic study of the multivariate complexity of
LCS, taking into account all parameters previously discussed in the literature:
the input size , the length of the shorter string
, the length of an LCS of and , the numbers of
deletions and , the alphabet size, as well as
the numbers of matching pairs and dominant pairs . For any class of
instances defined by fixing each parameter individually to a polynomial in
terms of the input size, we prove a SETH-based lower bound matching one of
three known algorithms. Specifically, we determine the optimal running time for
LCS under SETH as .
[...]Comment: Presented at SODA'18. Full Version. 66 page
Succinct Permutation Graphs
We present a succinct, i.e., asymptotically space-optimal, data structure for permutation graphs that supports distance, adjacency, neighborhood and shortest-path queries in optimal time; a variant of our data structure also supports degree queries in time independent of the neighborhood's size at the expense of an -factor overhead in all running times. We show how to generalize our data structure to the class of circular permutation graphs with asymptotically no extra space, while supporting the same queries in optimal time. Furthermore, we develop a similar compact data structure for the special case of bipartite permutation graphs and conjecture that it is succinct for this class. We demonstrate how to execute algorithms directly over our succinct representations for several combinatorial problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Moreover, we initiate the study of semi-local graph representations; a concept that "interpolates" between local labeling schemes and standard "centralized" data structures. We show how to turn some of our data structures into semi-local representations by storing only bits of additional global information, beating the lower bound on distance labeling schemes for permutation graphs
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