First-Fit Coloring of Incomparability Graphs

Bosek, Bartłomiej; Krawczyk, Tomasz; Matecki, Grzegorz

doi:10.1137/110854394

Cited by 5 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This problem seems structurally similar to the problem of 3colouring a graph formed by a union of three incomparability graphs. However, existing results on incomparability graphs (see, e.g., Bosek, Krawczyk, and Matecki 2013) do not seem to be directly applicable to our problem.…”

Section: Voter Partitionmentioning

confidence: 85%

On Recognising Nearly Single-Crossing Preferences

Jaeckle¹,

Peters

Elkind

2018

AAAI

View full text Add to dashboard Cite

If voters' preferences are one-dimensional, many hard problems in computational social choice become tractable. A preference profile can be classified as one-dimensional if it has the single-crossing property, which requires that the voters can be ordered from left to right so that their preferences are consistent with this order. In practice, preferences may exhibit some one-dimensional structure, despite not being single-crossing in the formal sense. Hence, we ask whether one can identify preference profiles that are close to being single-crossing. We consider three distance measures, which are based on partitioning voters or candidates or performing a small number of swaps in each vote. We prove that it can be efficiently decided if voters can be split into two single-crossing groups. Also, for every fixed k >= 1 we can decide in polynomial time if a profile can be made single-crossing by performing at most k candidate swaps per vote. In contrast, for each k >= 3 it is NP-complete to decide whether candidates can be partitioned into k sets so that the restriction of the input profile to each set is single-crossing.

show abstract

Section: Voter Partitionmentioning

confidence: 85%

On Recognising Nearly Single-Crossing Preferences

Jaeckle¹,

Peters

Elkind

2018

AAAI

View full text Add to dashboard Cite

show abstract

“…A poset constructed during a regular game of width w is L 2w 2 −ladder-free, see Figure 19. As ladders are of width 2, we deduce that First-Fit uses bounded number of chains in a regular game of width w. However, bounding functions proposed in [6] are not polynomial. Therefore, it would be desirable to answer the following questions.…”

Section: Open Problemsmentioning

confidence: 93%

“…In [15] Kierstead showed a countable poset of width 2 on which First-Fit uses infinitely many chains. In [6] Bosek, Krawczyk and Matecki noted that this poset contains all finite posets of width 2. They also showed that for every finite poset Q of width 2 there exists a function f Q : N → N such that FirstFit uses at most f Q (w) chains on Q-free posets of width w (a poset P is Q-free if it does not contain an induced poset isomorphic to Q).…”

Section: Open Problemsmentioning

confidence: 99%

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

Bosek

Krawczyk

2014

Combinatorica

Self Cite

View full text Add to dashboard Cite

The main question in the on-line chain partitioning problem is to decide whether there exists an on-line algorithm that partitions posets of width at most w into polynomial number of chains -see Trotter's chapter Partially ordered sets in the Handbook of Combinatorics. So far the best known on-line algorithm of Kierstead used at most (5 w − 1)/4 chains; on the other hand Szemerédi proved that any on-line algorithm requires at least w+1 2 chains. These results were obtained in the early eighties and since then no progress in the general case has been done.We provide an on-line algorithm that partitions posets of width w into at most w 13 log w chains. This yields the first subexponential upper bound for the on-line chain partitioning problem.

show abstract

“…[2]). For every poset Q of width 2, there is a function f Q such that every poset P of width w that does not contain Q as an induced subposet satisfies Γ (P) f Q (w).…”

Section: Conjecturementioning

confidence: 99%

Asymmetric Coloring Games on Incomparability Graphs

Krawczyk

Walczak

2015

Electronic Notes in Discrete Mathematics

Self Cite

View full text Add to dashboard Cite

Abstract. Consider the following game on a graph G: Alice and Bob take turns coloring the vertices of G properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of G is the minimum number of colors that allows Alice to win the game. The game Grundy number of G is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The (a, b)-game chromatic and Grundy numbers are defined likewise except that Alice colors a vertices and Bob colors b vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs (a, b) for which the (a, b)-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu.

show abstract

First-Fit Coloring of Incomparability Graphs

Cited by 5 publications

References 23 publications

On Recognising Nearly Single-Crossing Preferences

On Recognising Nearly Single-Crossing Preferences

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

Asymmetric Coloring Games on Incomparability Graphs

Contact Info

Product

Resources

About