A subexponential upper bound for the on-line chain partitioning problem

Bosek, Bartłomiej; Krawczyk, Tomasz

doi:10.1007/s00493-014-2908-7

Cited by 4 publications

(2 citation statements)

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“…Thus we replace Dilworth's theorem by Kierstead's effective analog, which states that every recursive partial order of width ≤ k can be decomposed into at most (5 k -1)/4 recursive chains [18]. Nowadays much better sub-exponential bounds are known for the number of recursive chains into which a recursive partial order of finite width can be decomposed [3].…”

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confidence: 99%

(Extra)ordinary Equivalences With the Ascending/Descending Sequence Principle

Fiori-Carones¹,

Marcone²,

Shafer³

et al. 2022

J. symb. log.

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We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival–Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ over $\mathsf {RCA}_0$ . For each fixed $k \geq 3$ , the Rival–Sands theorem for infinite partial orders of width $\leq \!k$ is equivalent to $\mathsf {ADS}$ over $\mathsf {RCA}_0$ . The Rival–Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf {SADS}$ over $\mathsf {RCA}_0$ . Here $\mathsf {RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf {I}\Sigma ^0_{2}$ denotes the $\Sigma ^0_2$ induction scheme, $\mathsf {ADS}$ denotes the ascending/descending sequence principle, and $\mathsf {SADS}$ denotes the stable ascending/descending sequence principle. To the best of 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 $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ , by $\mathsf {ADS}$ , and by $\mathsf {SADS}$ . 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 $\mathsf {RCA}_0$ .

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confidence: 99%

(Extra)ordinary Equivalences With the Ascending/Descending Sequence Principle

Fiori-Carones¹,

Marcone²,

Shafer³

et al. 2022

J. symb. log.

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show abstract

“…The first was found by Bosek and Krawczyk 2010 [40] (𝑤 16log 𝑤 many chains). Later this bound was improved by Bosek and Krawczyk 2015 [49] to 𝑤 13log 𝑤 many chains. Then again by Bosek et al 2018 [50] to 𝑤 6.5log 𝑤 + 7 .…”

Section: Online Partial Ordersmentioning

confidence: 99%

Adversarial and Online Algorithms

Askes

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In this thesis we explore a variety of online and adversarial algorithms. We primarily explore the following online and adversarial algorithms; the perfect code game, adversarial online colouring, the chain decomposition game, and strongly online graphs. The perfect code game is a new adversarial game played on graphs in which players take turns constructing perfect codes. We provide definitions for perfect codes and the perfect code game, along with some motivation from coding theory. We will prove upper bounds for both cycle and path graphs. We also prove an upper bound for graphs of bounded pathwidth. Finally, we explore the perfect code game in graphs of bounded degree. We will create a new game called the adversarial online colouring game, this game takes elements from both online and adversarial algorithms. We will begin with some discussion and a definition of adversarial online colouring. We will then prove several results related to graph degree. We conclude with a proof that the adversarial online colouring game on trees is determined only by the number of colours and the number of vertices (assuming that both players have a chance at winning). The chain decomposition game is another adversarial game, but this time played on partial orders. We introduce the chain decomposition game and demonstrate two results relating to upper and lower bounds for the game. We also prove a result on the online adversarial version of the game. We introduce strongly online graphs and graph colouring as a new algorithmic parameterization to online graph colouring. A strongly online graph is an online graph where at each stage 𝑠 we can see a ball of increasing radius about each vertex. We will prove several bounds (upper and lower) on the online chromatic number of strongly online graphs. For example, we show that every strongly online graph can be coloured in twice its chromatic number. We prove that every strongly online graph with even pathwidth 𝑘 can be online coloured with 2𝑘 + 1 colours. Then, after introducing a natural notion of strongly online pathwidth, we prove that there is a strongly online graph with no finite strongly online path decomposition.

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Improved lower bound on the on-line chain partitioning of semi-orders with representation

Biró,

Curbelo

2023

Discrete Mathematics

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A subexponential upper bound for the on-line chain partitioning problem

Cited by 4 publications

References 12 publications

(Extra)ordinary Equivalences With the Ascending/Descending Sequence Principle

(Extra)ordinary Equivalences With the Ascending/Descending Sequence Principle

Adversarial and Online Algorithms

Improved lower bound on the on-line chain partitioning of semi-orders with representation

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