Computing and Combinatorics: 17th Annual International by Kazuhisa Makino, Suguru Tamaki, Masaki Yamamoto (auth.), Bin

By Kazuhisa Makino, Suguru Tamaki, Masaki Yamamoto (auth.), Bin Fu, Ding-Zhu Du (eds.)

This e-book constitutes the refereed court cases of the sixteenth Annual overseas convention on Computing and Combinatorics, held in Dallas, TX, united states, in August 2011. The fifty four revised complete papers provided have been rigorously reviewed and chosen from 136 submissions. themes lined are algorithms and knowledge buildings; algorithmic online game idea and on-line algorithms; automata, languages, common sense, and computability; combinatorics relating to algorithms and complexity; complexity idea; computational studying conception and information discovery; cryptography, reliability and protection, and database thought; computational biology and bioinformatics; computational algebra, geometry, and quantity concept; graph drawing and data visualization; graph thought, verbal exchange networks, and optimization; parallel and dispensed computing.

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Additional info for Computing and Combinatorics: 17th Annual International Conference, COCOON 2011, Dallas, TX, USA, August 14-16, 2011. Proceedings

Example text

To compute an approximately optimal solution for bucket i we structure the input by considering instances Ii = (G, i , wi , Wi , Li ), where (e)/ki , wi (e) = w(e)/k i for e ∈ E(G) and Wi = W/k i as well as i (e) = Li = L/ki. We apply algorithm A on instance Ii with λ = k 2 m. When considering the i-th bucket, we refer to the deviation of H ⊆ G with respect to i and Li as Δi (H) = max{0, i(H) − Li }. Similarly the penalized density of H ⊆ G is defined as i (H) = wi (H)/( i (H)+Δi (H)). Lemma 3 shows that our structuring scheme implies that i (H) is monotonically decreasing in i for each H ⊆ G.

Proof. Clearly, this inequality holds when Ω(H) = 0. For Ω(H) ≥ 1, by Lemma 3, Equation 1, and |E(H)| ≤ m we get (H) ≥ k Ω(H) · ( Ω(H) (H) − m). Together with Corollary 2 this implies k Ω(H) m ≤ 1/(k − 1) · (H) ≤ 1/(k − 1) · ( (H) + Δ(H)). We now compute the density of H in Iteration ω := Ω(H) by using Equations (1) and (2). wω (H) w(H) = ω k · ( ω (H) + Δω (H)) ω (H) + Δω (H) wt(H) w(H) k−1 ≥ ≥ = · (H) . 2 (H) + Δ(H) + 2m · k ω k+1 1 + k−1 ( (H) + Δ(H)) ω (H) = The Density Maximization Problem in Graphs 35 Theorem 5.

J}. For an edge e = {u, v} ∈ E such that u ∈ X or v ∈ Y , at least one of pei and qie should belong to D for every i ∈ {1, . . , i}. Therefore, for fixed X ⊆ U and Y ⊆ V , the number of dominating sets D such that X = D ∩ U and Y = D ∩ V is equal to 2j|X| · 2j|Y | · 22i {u,v}∈E, u∈X,v∈Y 3i = 2j(|X|+|Y |) · 3i|E| · {u,v}∈E, u∈X or v∈Y 4 3 i·e(X,Y ) , where e(X, Y ) = |{{u, v} ∈ E | u ∈ X, v ∈ Y }|. Therefore, the number of dominating sets in Gi,j is equal to 2j(|X|+|Y |) · 3i|E| · dom(Gi,j ) = X⊆U Y ⊆V |E| = k=0 ⎛ ⎞ ⎜ ⎜ ⎝ j(|X|+|Y |) ⎟ 2 4 3 i·e(X,Y ) ⎟ |E| ⎠ 3 X⊆U,Y ⊆V, e(X,Y )=k 4 3 k i .

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