6 research outputs found
An easy subexponential bound for online chain partitioning
Bosek and Krawczyk exhibited an online algorithm for partitioning an online
poset of width into chains. We improve this to with a simpler and shorter proof by combining the work of Bosek &
Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of
ladder-free posets. We also provide examples illustrating the limits of our
approach.Comment: 23 pages, 11 figure
On-line partitioning of width w posets into w^O(log log w) chains
An on-line chain partitioning algorithm receives the elements of a poset one
at a time, and when an element is received, irrevocably assigns it to one of
the chains. In this paper, we present an on-line algorithm that partitions
posets of width into chains. This improves over
previously best known algorithms using chains by Bosek and
Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm
runs in time, where is the width and is the size of
a presented poset.Comment: 16 pages, 10 figure
Recoloring Interval Graphs with Limited Recourse Budget
We consider the problem of coloring an interval graph dynamically. Intervals arrive one after the other and have to be colored immediately such that no two intervals of the same color overlap. In each step only a limited number of intervals may be recolored to maintain a proper coloring (thus interpolating between the well-studied online and offline settings). The number of allowed recolorings per step is the so-called recourse budget. Our main aim is to prove both upper and lower bounds on the required recourse budget for interval graphs, given a bound on the allowed number of colors. For general interval graphs with n vertices and chromatic number k it is known that some recoloring is needed even if we have 2k colors available. We give an algorithm that maintains a 2k-coloring with an amortized recourse budget of . For maintaining a k-coloring with k ≤ n, we give an amortized upper bound of \u1d4aa(k⋅ k! ⋅ √n), and a lower bound of , which can be as large as ). For unit interval graphs it is known that some recoloring is needed even if we have k+1 colors available. We give an algorithm that maintains a (k+1)-coloring with at most recolorings per step in the worst case. We also give a lower bound of on the amortized recourse budget needed to maintain a k-coloring. Additionally, for general interval graphs we show that if one does not insist on maintaining an explicit coloring, one can have a k-coloring algorithm which does not incur a factor of in the running time. For this we provide a data structure, which allows for adding intervals in amortized time per update and querying for the color of a particular interval in . Between any two updates, the data structure answers consistently with some optimal coloring. The data structure maintains the coloring implicitly, so the notion of recourse budget does not apply to it
The on-line width of various classes of posets.
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\\u27edi proved that any on-line algorithm could be forced to use chains to partition a poset of width . The maximum number of chains that can be forced on any on-line algorithm remains unknown. In the survey paper by Bosek et al., variants of the problem were studied where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size . We prove two results for this problem. First, we prove that any on-line algorithm can be forced to use chains to partition a -dimensional poset of width . Second, we prove that any on-line algorithm can be forced to use chains to partition a poset of width presented via a realizer of size . Chrobak and \\u27Slusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most chains, where is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most chains, over years later, it remains unknown whether a more optimal algorithm exists. We improve upon previously known bounds and show that any on-line algorithm can be forced to use chains to partition a semi-order presented in the form of its unit-interval representation. As a consequence, we completely solve the problem for . We also consider entirely new variants and present the results for those
(Extra)ordinary equivalences with the ascending/descending sequence principle
We analyze the axiomatic strength of the following theorem due to Rival and
Sands in the style of reverse mathematics. "Every infinite partial order of
finite width contains an infinite chain such that every element of is
either comparable with no element of or with infinitely many elements of
." Our main results are the following. The Rival-Sands theorem for infinite
partial orders of arbitrary finite width is equivalent to over . For each fixed , the
Rival-Sands theorem for infinite partial orders of width is
equivalent to over . The Rival-Sands theorem for
infinite partial orders that are decomposable into the union of two chains is
equivalent to over . Here
denotes the recursive comprehension axiomatic system,
denotes the induction scheme, denotes the
ascending/descending sequence principle, and denotes the stable
ascending/descending sequence principle. To our knowledge, these versions of
the Rival-Sands theorem for partial orders are the first examples of theorems
from the general mathematics literature whose strength is exactly characterized
by , by , and by
. Furthermore, we give a new purely combinatorial result by
extending the Rival-Sands theorem to infinite partial orders that do not have
infinite antichains, and we show that this extension is equivalent to
arithmetical comprehension over
Filling cages: reverse mathematics and combinatorial principles
Nella tesi sono analizzati alcuni principi di combinatorica dal punto di vista della reverse mathematics. La reverse mathematics \ue8 un programma di ricerca avviato negli anni settanta e interessato a individuare l'esatta forza, intesa come assiomi riguardanti l'esistenza di insiemi, di teoremi della matematica ordinaria. ---
Dopo una concisa introduzione al tema, \ue8 presentato un algoritmo incrementale per reorientare transitivamente grafi orientati infiniti e pseudo-transitivi. L'esistenza di tale algoritmo implica che un teorema di Ghouila-Houri \ue8 dimostrabile in RCA0. ---
Grafi e ordini a intervalli sono la comune tematica della seconda parte della tesi. Un primo capitolo \ue8 dedicato all'analisi di diverse caratterizzazioni di grafi numerabili a intervalli e allo studio della relazione tra grafi numerabili a intervalli e ordini numerabili a intervalli. In questo contesto emerge il tema dell'ordinabilit\ue0 unica di grafi a intervalli, a cui \ue8 dedicato il capitolo successivo. L'ultimo capitolo di questa parte riguarda invece enunciati relativi alla dimensione degli ordini numerabili a intervalli. ---
La terza parte ruota attorno due enunciati dimostrati da Rival e Sands in un articolo del 1980. Il primo teorema afferma che ogni grafo infinito contiene un sottografo infinito tale che ogni vertice del grafo \ue8 adiacente ad al pi\uf9 uno o a infiniti vertici del sottografo. Si dimostra che questo enunciato \ue8 equivalente ad ACA0, dunque pi\uf9 forte rispetto al teorema di Ramsey per coppie, nonostante la somiglianza dei due principi. Il secondo teorema dimostrato da Rival e Sands asserisce che ogni ordine parziale infinito con larghezza finita contiene una catena infinita tale che ogni punto dell'ordine \ue8 comparabile con nessuno o con infiniti elementi della catena. Quest'ultimo enunciato ristretto a ordini di larghezza k, per ogni k maggiore o uguale a tre, \ue8 dimostrato equivalente ad ADS. Ulteriori enunciati sono studiati nella tesi.In the thesis some combinatorial statements are analysed from the reverse mathematics point of view. Reverse mathematics is a research program, which dates back to the Seventies, interested to find the exact strength, measured in terms of set-existence axioms, of theorems from ordinary non set-theoretic mathematics. ---
After a brief introduction to the subject, an on-line (incremental) algorithm to transitivelly reorient infinite pseudo-transitive oriented graphs is defined. This implies that a theorem of Ghouila-Houri is provable in RCA0 and hence is computably true. ---
Interval graphs and interval orders are the common theme of the second part of the thesis. A chapter is devoted to analyse the relative strength of different characterisations of countable interval graphs and to study the interplay between countable interval graphs and countable interval orders. In this context arises the theme of unique orderability of interval graphs, which is studied in the following chapter. The last chapter about interval orders inspects the strength of some statements involving the dimension of countable interval orders. ---
The third part is devoted to the analysis of two theorems proved by Rival and Sands in 1980. The first principle states that each infinite graph contains an infinite subgraph such that each vertex of the graph is adjacent either to none, or to one or to infinitely many vertices of the subgraph. This statement, restricted to countable graphs, is proved to be equivalent to ACA0 and hence to be stronger than Ramsey's theorem for pairs, despite the similarity of the two principles. The second theorem proved by Rival and Sands states that each infinite partial order with finite width contains an infinite chain such that each point of the poset is comparable either to none or to infinitely many points of the chain. For each k greater or equal to three, the latter principle restricted to countable poset of width k is proved to be equivalent to ADS. Some complementary results are presented in the thesis