Computing and Combinatorics: 18th Annual International by Binay Bhattacharya, Tsunehiko Kameda (auth.), Joachim

By Binay Bhattacharya, Tsunehiko Kameda (auth.), Joachim Gudmundsson, Julián Mestre, Taso Viglas (eds.)

This booklet constitutes the refereed court cases of the 18th Annual foreign convention on Computing and Combinatorics, held in Sydney, Australia, in August 2012.
The 50 revised complete papers offered have been rigorously reviewed and chosen from 121 submissions. subject matters lined are algorithms and knowledge constructions; algorithmic video game conception and on-line algorithms; automata, languages, good judgment, and computability; combinatorics with regards to algorithms and complexity; complexity conception; computational studying idea and information discovery; cryptography, reliability and safeguard, and database conception; computational biology and bioinformatics; computational algebra, geometry, and quantity idea; graph drawing and knowledge visualization; graph conception, conversation networks, and optimization.

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Lemma 5. If 0 < ≤ 1/2, then l∈I / k l=1 Δ(Ol ,C (i−1) ) Δ(Ol ,C (i−1) ) ≥ /2. 20 R. Jaiswal, A. Kumar, and S. Sen Proof. Suppose, for the sake of contradiction, the above statement does not hold. Then, Δ(P, C (i−1) ) = Δ(Ol , C (i−1) ) + l∈I / l∈I Δ(Ol , C (i−1) ) + < l∈I = ≤ Δ(Ol , C (i−1) ) 1 · 1 − /2 /2 · 1 − /2 Δ(Ol , C (i−1) ) (by our assumption) l∈I Δ(Ol , C (i−1) ) l∈I 1 + /20 · 1 − /2 (using the invariant for C (i−1) ) Δ1 (Ol ) l∈I ≤ (1 + ) · Δ1 (Ol ) (since ≤ 1/2) l∈I ≤ (1 + ) · Δ1 (Ol ) l∈[k] But this contradicts the fact that P is (k, )-irreducible.

In the context of our conjecture, let us recall that the problems of detecting an independent set on k vertices and detecting a clique on k vertices in a host graph on n vertices are known to be W[1]-hard in the theory of parametrized complexity and believed to require nΩ(k) time [5]. Known Results Supporting the Conjecture. Already in 1985, Ne˘set˘ril and Poljak showed in [14] that the detection and counting versions of the induced subgraph isomorphism with fixed pattern graph on k vertices are easily reducible to the corresponding versions of the k-clique problem (or, equivalently, the kindependent set problem) in O(kn2 ) time, where n is the number of vertices of the host graph.

Suppose that there is a maximum independent set of size k that is disjoint from all other maximum independent sets in H. In O(kn2 +hkn+k 2m) time, one can construct a graph G∗ on O(h+kn) vertices 42 P. Floderus et al. and O(kn2 + hkn + k 2m) edges such that G∗ has an induced subgraph isomorphic to H iff G has an independent set of cardinality k. Proof. Let G = (V, E) and H = (VH , EH ). Next, let S be a maximum independent set that is disjoint from the other maximum independent sets in H. G∗ consists of k = |S| cliques G∗ (i) on V × {i}, where i ∈ S, and the subgraph H of H induced by all vertices in VH outside S.

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