The complexity of Shortest Common Supersequence for inputs with no identical consecutive letters

The Shortest Common Supersequence problem (SCS for short) consists in finding a shortest common supersequence of a finite set of words on a fixed alphabet Σ. It is well-known that its decision version denoted [SR8] in [3] is NP-complete. Many variants have been studied in the literature. In this paper we settle the complexity of two such variants of SCS where inputs do not contain identical consecutive letters. We prove that those variants denoted φSCS and MSCS both have a decision version which remains NP-complete when |Σ| ≥ 3. Note that it was known for MSCS when |Σ| ≥ 4 [2] and we discuss how [1] states a similar result for |Σ| ≥ 3. The two problems we study are precisely described below. In this paper, we will use the following terminology. Given two words over an alphabet Σ, u = u1 . . . up (ui ∈ Σ) and v = v1 . . . vq (vi ∈ Σ), an embedding of u into v is an injection f from {1, . . . , p} into {1, . . . , q} such that ui = vf(i). It tells that v is a supersequence of u and we also say that f maps letters of u onto letters of v. We will also use equivalently the terms pattern, block or factor to designate a sequence of consecutive letters in a word. A supersequence for a set of words is a word which is a supersequence for each of those words. To define the first problem, we use the notations Σ2 = {0, 1} and Σ3 = {0, 1, 2}, and we define the word morphism φ : Σ2 → Σ3 by φ(0) = 0202 and φ(1) = 1. Shortest Common Supersequence for some inputs generated by φ decision version φSCS Input: A set L = {w1, . . . , wn} of words on the alphabet Σ3 such that L ⊆ φ(Σ∗), each wi contains exactly two ones, which moreover are non consecutive, and an integer k. Output: Does there exist a supersequence of L of size less than k? The second variant has been named Modified SCS in [2]. Modified Shortest Common Supersequence (MSCS) decision version Input: A set L = {w1, . . . , wn} of words on an alphabet Σ = {a1, a2, . . . , ad} such that no word wi contains two consecutive identical letters and no word wi starts with letter a1, and an integer k. Output: Does there exist a supersequence of L of size less than k? A careful look at those two problems shows that φSCS is a particular case of MSCS if |Σ| ≥ 3. The input words for φSCS are a concatenation of patterns 0202 and 1 with no consecutive ones, thus they do not contain consecutive identical letters. Moreover none of those input words starts ∗LIP, Université de Lyon, ENS Lyon, CNRS UMR5668, INRIA, UCB Lyon 1, France