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A NOTE ON THE HARDNESS OF SKOLEMTYPE SEQUENCES
"... Abstract. The purpose of this note is to give upper bounds (assuming P different from NP) on how far the generalizations of Skolem sequences can be taken while still hoping to resolve the existence question. We prove that the existence questions for both multi Skolem sequences and generalized Skolem ..."
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Abstract. The purpose of this note is to give upper bounds (assuming P different from NP) on how far the generalizations of Skolem sequences can be taken while still hoping to resolve the existence question. We prove that the existence questions for both multi Skolem sequences and generalized Skolem sequences are strongly NPcomplete. These results are significant strengthenings and simplifications of the recent NPcompleteness result for generalized multi Skolem sequences. 1.
ITuniversity of Copenhagen Contents
, 2009
"... 1.1 Historical background........................ 1 1.2 Definitions and notation....................... 3 ..."
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1.1 Historical background........................ 1 1.2 Definitions and notation....................... 3
The Existence of Two New Classes of NearSkolem Type Sequences
"... Let m,n be two positive integers, m ≤ n. An mnearSkolem sequence of order n and defect m is a sequence m − Sn = (s1, s2, · · · , s2n−2) of 2n − 2 nonnegative integers such that the following conditions hold: 1. for each k ∈ {1, 2, · · ·,m−1,m+1, · · · , n} there exist exactly two element ..."
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Let m,n be two positive integers, m ≤ n. An mnearSkolem sequence of order n and defect m is a sequence m − Sn = (s1, s2, · · · , s2n−2) of 2n − 2 nonnegative integers such that the following conditions hold: 1. for each k ∈ {1, 2, · · ·,m−1,m+1, · · · , n} there exist exactly two elements si, sj ∈ m − Sn such that si = sj = k 2. if si = sj = k, then i − j  = k For example, (4, 1, 1, 2, 4, 2) is an example of a 3nearSkolem sequence of order 4. In this article we prove that the necessary conditions are also sufficient for the existence of two new classes of nearSkolem type sequences; one with a hook two places from the last entry of the sequence and one with a hook two places and one place from the last entry. The sequences