Computing and Combinatorics: 16th Annual International by Manuel Blum (auth.), My T. Thai, Sartaj Sahni (eds.)

By Manuel Blum (auth.), My T. Thai, Sartaj Sahni (eds.)

This ebook constitutes the complaints of the sixteenth Annual overseas convention on Computing and Combinatorics, held in Nha Trang, Vietnam, in July 2010.

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Additional info for Computing and Combinatorics: 16th Annual International Conference, COCOON 2010, Nha Trang, Vietnam, July 19-21, 2010. Proceedings

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More specifically, for each integer t ≥ 1 we define variants of the two problems, named t-total vertex cover and t-total edge cover. In t-total vertex cover, or t-tvc for short, (resp. t-total edge cover, or t-tec for short), we ask whether there is a vertex cover (resp. edge cover) S of the input graph such that each connected component of the subgraph induced on S contains at least t vertices (resp. at least t edges from S). These problems were introduced by Fernau and Manlove [12] who initiated the study of the parameterized complexity of these problems.

There exists a coordinate im such that Pr[Wim |W ] ≤ ω ♦ (G) + 13 1 log n−m 1 Pr[W ] Proof: Assume without loss of generality that i1 , . . , im−1 ∈ {n − m + 1, . . , n} (the last m coordinates). 2 we obtain: n−m PX n An Yi Zi |W − PX n An |W PYi Zi |X n An ≤ (n − m) log i=1 1 . Pr[W ] Since the strategy is No-Signaling: n−m PX n An Yi Zi |W − PX n An |W PZi Yi |Xi ≤ (n − m) log i 1 . Pr[W ] This can be written as: n−m PX n An Yi Zi |W − PXi |W PX −i An |Xi ,W PZi Yi |Xi ≤ (n − m) log i 1 . 4, there exists a distribution QX −i An Y −i B n Y −i C n |Xi =x,Yi =y,Zi =z which can be implemented by a No-Signaling function and satisfy: n−m PXY Z QX −i An Y −i B n Y −i C n |XY Z − PXY ZX −i An Y −i B n Y −i C n |W ≤ 13 (n − m) log i=1 1 .

Let P and Q be parameterized problems. We say that P is polynomial time and parameter reducible to Q, written P ≤P tp Q, if there exists a polynomial time computable function f : Σ ∗ × N → Σ ∗ × N, and a polynomial p : N → N, and for all x ∈ Σ ∗ and k ∈ N, if f ((x, k)) = (x , k ), then (x, k) ∈ P if and only if (x , k ) ∈ Q, and k ≤ p (k). We call f a polynomial parameter transformation (or a PPT) from P to Q. This notion of a reduction is useful in showing kernel lower bounds because of the following theorem: Fact 2.

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