Deciding the On-line Chromatic Number of a Graph with Pre-coloring Is PSPACE-Complete

Kudahl, Christian

doi:10.1007/978-3-319-18173-8_23

Cited by 4 publications

(4 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It remains open whether these problems are NP-complete, PSPACE-complete, or neither. In [16] it was shown that determining the online chromatic number is PSPACEcomplete if the graph is pre-colored and extended in [3] to hold even if the graph is not pre-colored.…”

Section: Complexity Of Determining the Online Independence Number Ver...mentioning

confidence: 99%

Adding isolated vertices makes some greedy online algorithms optimal

Boyar

Kudahl

2018

Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

An unexpected difference between online and offline algorithms is observed. The natural greedy algorithms are shown to be worst case online optimal for Online Independent Set and Online Vertex Cover on graphs with "enough" isolated vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is shown to be worst case online optimal on graphs with at least one isolated vertex. These algorithms are not online optimal in general. The online optimality results for these greedy algorithms imply optimality according to various worst case performance measures, such as the competitive ratio. It is also shown that, despite this worst case optimality, there are Freckle graphs where the greedy independent set algorithm is objectively less good than another algorithm.It is shown that it is NP-hard to determine any of the following for a given graph: the online independence number, the online vertex cover number, and the online domination number.

show abstract

Section: Complexity Of Determining the Online Independence Number Ver...mentioning

confidence: 99%

Adding isolated vertices makes some greedy online algorithms optimal

Boyar

Kudahl

2018

Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…We call this decision problem Online Chromatic Number with Precoloring. The paper [13] conjectures that Online Chromatic Number (with no precolored part) is PSPACE-complete too. Interestingly, it is possible to decide χ O (G) ≤ 3 in polynomial time [7].…”

Section: Definition 2 the Online Chromatic Number Problem Is As Follmentioning

confidence: 99%

“…Inspired by [13], we prove the PSPACE-hardness of Online Chromatic Number by a reduction from Q3DNF-SAT, i.e., the satisfiability of a fully quantified formula in the 3-disjunctive normal form (3-DNF). An example of such a formula is…”

Section: Definition 2 the Online Chromatic Number Problem Is As Follmentioning

confidence: 99%

“…We note that if we would give Painter some advantage like parallel edges, precolored vertices (as in [13]) or something that helps Painter distinguish different parts of the graph, the proof of PSPACE-hardness would be much simpler. However, our theorem holds for simple graphs without any such help.…”

Section: Definition 2 the Online Chromatic Number Problem Is As Follmentioning

confidence: 99%

See 1 more Smart Citation

Online Chromatic Number is PSPACE-Complete

Böhm

Veselý

2016

Lecture Notes in Computer Science

View full text Add to dashboard Cite

In the online graph coloring problem, vertices from a graph G, known in advance, arrive in an online fashion and an algorithm must immediately assign a color to each incoming vertex v so that the revealed graph is properly colored. The exact location of v in the graph G is not known to the algorithm, since it sees only previously colored neighbors of v. The online chromatic number of G is the smallest number of colors such that some online algorithm is able to properly color G for any incoming order. We prove that computing the online chromatic number of a graph is PSPACE-complete.

show abstract