Approximability Distance in the Space of H-Colourability Problems

Färnqvist, Tommy; Jönsson, Peter; Thapper, Johan

doi:10.1007/978-3-642-03351-3_11

Cited by 3 publications

(5 citation statements)

References 27 publications

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“…We choose to begin our study of s(K 2 , K t ) using small values of n. When n = 1, K 2+1/k is isomorphic to the cycle C 2k+1 . The value of s(K 2 , C 2k+1 ) = 2k/(2k + 1), for k ≥ 1 was obtained in [2]. Combined with the following result, where we set t = 2 + 2/(2k − 1) = 4k 2k−1 , this has an immediate and perhaps surprising consequence.…”

Section: Maps To Kmentioning

confidence: 59%

“…Since MAX H -COL is a hard problem to solve exactly, efforts have been made to find suitable approximation algorithms. Färnqvist et al [2] introduce a method that can be used to extend previously known (in)approximability bounds on MAX H -COL to new and larger classes of graphs. For example, they present concrete approximation ratios for certain graphs (such as the odd cycles) and near-optimal asymptotic results for large graph classes.…”

Section: Measurementioning

confidence: 99%

“…Färnqvist et al [2] have identified an alternative expression for s(M, N ) which depends on the automorphism group of N . Let M and N ∈ G be graphs and let A = Aut * (N ) be the (edge) automorphism group of N , i.e., π ∈ A acts on E(N ) by permuting the edges.…”

Section: A Linear Program For Smentioning

confidence: 99%

“…Similar intervals appear throughout the space of circular complete graphs. More specifically, Färnqvist et al [2] As we proceed with determining s(K 2 , K t ) we can now, thanks to Corollary 3.2, disregard those t which fall inside these constant intervals. For t = 2 + 3/k, we see that if k ≡ 0 (mod 3), then r is an odd cycle, and if k ≡ 2 (mod 3), then t ∈ I k+1 .…”

Section: Maps To Kmentioning

confidence: 99%

“…However, it is notoriously difficult to find out exactly how well the algorithm approximates MAX CSP(Γ) for a given Γ. It seems plausible that the function s can be extended into a function s ′ from pairs of sets of relations to Q + , and that s ′ can be used for studying the approximability of MAX CSP by extending the approach in Färnqvist et al [2]. This would constitute a novel method for studying the approximability of MAX CSP -a method that, hopefully, may cast some new light on the performance of Raghavendra's algorithm.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

See 4 more Smart Citations

Graph Homomorphisms, Circular Colouring, and Fractional Covering by H-cuts

Engström,

Färnqvist,

Jonsson

et al. 2009

Preprint

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A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H -COL) are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H = K k this problem is equivalent to MAX k-CUT. Färnqvist et al. have introduced a parameter on the space of graphs that allows close study of the approximability properties of MAX H -COL. Specifically, it can be used to extend previously known (in)approximability results to larger classes of graphs. Here, we investigate the properties of this parameter on circular complete graphs K p/q , where 2 ≤ p/q ≤ 3. The results are extended to K 4 -minor-free graphs and graphs with bounded maximum average degree. We also consider connections with Šámal's work on fractional covering by cuts: we address, and decide, two conjectures concerning cubical chromatic numbers.

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Section: Maps To Kmentioning

confidence: 59%

Section: Measurementioning

confidence: 99%

Section: A Linear Program For Smentioning

confidence: 99%

Section: Maps To Kmentioning

confidence: 99%

Section: Conclusion and Open Problemsmentioning

confidence: 99%

See 3 more Smart Citations

Graph Homomorphisms, Circular Colouring, and Fractional Covering by H-cuts

Engström,

Färnqvist,

Jonsson

et al. 2009

Preprint

Self Cite

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Properties of an Approximability-related Parameter on Circular Complete Graphs

Engström

Färnqvist

Jönsson

et al. 2009

Electronic Notes in Discrete Mathematics

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An approximability-related parameter on graphs―-properties and applications

Engström¹,

Färnqvist²,

Jönsson³

et al. 2015

Discrete Mathematics &Amp; Theoretical Computer Science

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Graph Theory International audience We introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col.

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Approximability Distance in the Space of H-Colourability Problems

Cited by 3 publications

References 27 publications

Graph Homomorphisms, Circular Colouring, and Fractional Covering by H-cuts

Graph Homomorphisms, Circular Colouring, and Fractional Covering by H-cuts

Properties of an Approximability-related Parameter on Circular Complete Graphs

An approximability-related parameter on graphs―-properties and applications

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