Optimal Approximation of Sparse Hessians and Its Equivalence to a Graph Coloring Problem.

McCormick, S. Thomas

doi:10.21236/ada110846

Cited by 39 publications

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“…Note that no results are known about the approximability of the L(h, k)-labelling problem on general graphs, while for the coloring of square graphs an O( √ n)-approximation algorithm exists [14]. In this context our result strongly relates the approximability of these two problems.…”

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confidence: 84%

“…It has been proven that the decision version of the L(h, k)-labelling problem is NP-complete even for h ∈ {1, 2} and k = 1 [14,13,12]. Therefore, the problem has been widely studied for many specific classes of graphs.…”

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confidence: 99%

“…It is easy to see that when h = k = 1, the L(h, k)-labelling problem on a graph G is equivalent to vertex-coloring G 2 , where G 2 is the square graph of G, and a wide literature is known in this case (e.g., see [3,14]). …”

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confidence: 99%

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On the Approximability of the L(h,k)-Labelling Problem on Bipartite Graphs (Extended Abstract)

Calamoneri

Vocca

2005

Structural Information and Communication Complexity

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Abstract. Given an undirected graph G, an L(h, k)-labelling of G assigns colors to vertices from the integer set {0, . . . , λ h,k }, such that any two vertices vi and vj receive colors c(vi) and c(vj ) satisfying the following conditions: i) if vi and vj are adjacent then |c(vi) − c(vj)| ≥ h; ii) if vi and vj are at distance two then |c(vi) − c(vj)| ≥ k. The aim of the L(h, k)-labelling problem is to minimize λ h,k . In this paper we study the approximability of the L(h, k)-labelling problem on bipartite graphs and extend the results to s-partite and general graphs. Indeed, the decision version of this problem is known to be NP-complete in general and, to our knowledge, there are no polynomial solutions, either exact or approximate, for bipartite graphs. Here, we state some results concerning the approximability of the L(h, k)-labelling problem for bipartite graphs, exploiting a novel technique, consisting in computing approximate vertex-and edge-colorings of auxiliary graphs to deduce an L(h, k)-labelling for the input bipartite graph. We derive an approximation algorithm with performance ratio bounded by 4 3 D 2 , where, D is equal to the minimum even value bounding the minimum of the maximum degrees of the two partitions. One of the above coloring algorithms is in fact an approximating edge-coloring algorithm for hypergraphs of maximum dimension d, i.e. the maximum edge cardinality, with performance ratio d. Furthermore, we consider a different approximation technique based on the reduction of the L(h, k)-labelling problem to the vertex-coloring of the square of a graph. Using this approach we derive an approximation algorithm with performance ratio bounded by min(h, 2k)Hence, the first technique is competitive when D = O(n 1/4 ). These algorithms match with a result in [2] stating that L(1, 1)-labelling n-vertex bipartite graphs is hard to approximate within n 1/2−ǫ , for any ǫ > 0, unless NP=ZPP. We then extend the latter approximation strategy to s-partite graphs, obtaining a (min(h, sk) √ n+ o(sk √ n))-approximation ratio, and to general graphs deriving an (h √ n+o(h √ n))-approximation algorithm, assuming h ≥ k. Finally, we prove that the L(h, k)-labelling problem is not easier than coloring the square of a graph.

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On the Approximability of the L(h,k)-Labelling Problem on Bipartite Graphs (Extended Abstract)

Calamoneri

Vocca

2005

Structural Information and Communication Complexity

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“…The Vizing's theorem, [1] states that the minimum number of colors needed to 1-hop color a graph denoted χ, meets ∆ χ ∆+1, where ∆ is the maximum degree of the graph. The h-hop node coloring problem has been proved NP-complete in [2] for h = 1 and in [3,4] for any h > 1. This explains why heuristics are used for large graphs.…”

Section: Introduction and State Of The Artmentioning

confidence: 99%

OSERENA: a Coloring Algorithm Optimized for Dense Wireless Networks

Amdouni¹,

Minet²,

Adjih³

2013

IJNDC

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The goal of this paper is to present OSERENA, a distributed coloring algorithm optimized for dense wireless sensor networks (WSNs). Network density has an extremely reduced impact on the size of the messages exchanged to color the WSN. Furthermore, the number of colors used to color the network is not impacted by this optimization. We describe in this paper the properties of the algorithm and prove its correctness and termination. Simulation results point out the considerable gains in bandwidth.

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“…. , 1), the channel assignment problem has been widely studied in the past [2,3,14,21,23]. In particular, the intractability of optimal L(1, .…”

Section: Introductionmentioning

confidence: 99%

Approximate L(δ₁,δ₂,…,δ_t)‐coloring of trees and interval graphs

Bertossi

Pinotti

2007

Networks

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Given a vector (δ 1 , δ 2 , . . . , δ t ) of nonincreasing positive integers, and an undirected graph G = (V , E ), an L(δ 1 , δ 2 , . . . , δ t )-coloring of G is a function f from the vertex set V to a set of nonnegative integers such thatis the distance (i.e., the minimum number of edges) between the vertices u and v . An optimal L(δ 1 , δ 2 , . . . , δ t )-coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(δ 1 , δ 2 , . . . , δ t )-coloring of two relevant classes of graphs-trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t +δ 1 )) and O(nt 2 δ 1 ) time algorithms are proposed to find α-approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and α is a constant depending on t and δ 1 , . . . , δ t . Moreover, an O(n) time algorithm is given for the L(δ 1 , δ 2 )-coloring of unit interval graphs, which provides a 3-approximation.

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Optimal Approximation of Sparse Hessians and Its Equivalence to a Graph Coloring Problem.

Cited by 39 publications

References 0 publications

On the Approximability of the L(h,k)-Labelling Problem on Bipartite Graphs (Extended Abstract)

On the Approximability of the L(h,k)-Labelling Problem on Bipartite Graphs (Extended Abstract)

OSERENA: a Coloring Algorithm Optimized for Dense Wireless Networks

Approximate L(δ₁,δ₂,…,δ_t)‐coloring of trees and interval graphs

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Optimal Approximation of Sparse Hessians and Its Equivalence to a Graph Coloring Problem.

Cited by 39 publications

References 0 publications

On the Approximability of the L(h,k)-Labelling Problem on Bipartite Graphs (Extended Abstract)

On the Approximability of the L(h,k)-Labelling Problem on Bipartite Graphs (Extended Abstract)

OSERENA: a Coloring Algorithm Optimized for Dense Wireless Networks

Approximate L(δ1,δ2,…,δt)‐coloring of trees and interval graphs

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Product

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Approximate L(δ₁,δ₂,…,δ_t)‐coloring of trees and interval graphs