Abstract. Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multiple-interval graphs was initiated. In this sequel, we study multiple-interval graph problems from the perspective of parameterized complexity. The problems under consideration are k-Independent Set, k-Dominating Set, and k-Clique, which are all known to be W[1]-hard for general graphs, and NP-complete for multiple-interval graphs. We prove that k-Clique is in FPT, while k-Independent Set and k-Dominating Set are both W[1]-hard. We also prove that k-Independent Dominating Set, a hybrid of the two above problems, is also W[1]-hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]-hardness via a reduction from the k-Multicolored Clique problem, a variant of k-Clique. We believe this technique has interest in its own right, as it should help in simplifying W[1]-hardness results which are notoriously hard to construct and technically tedious.

Abstract. Given two genomic maps G and H represented by a sequence of n gene markers, a strip (syntenic block) is a sequence of distinct markers of length at least two which appear as subsequences in the input maps, either directly or in reversed and negated form. The problem Maximal Strip Recovery (MSR) is to find two subsequences G ′ and H ′ of G and H, respectively, such that the total length of disjoint strips in G ′ and H ′ is maximized (or, conversely, the number of markers hence deleted, is minimized). Previously, besides some heuristic solutions, a factor-4 polynomial-time approximation is known for the MSR problem; moreover, several close variants of MSR, MSR-d (with d>2 input maps), MSR-DU (with marker duplications) and MSR-WT (with markers weighted) are all shown to be NP-complete. Before this work, the complexity of the original MSR problem was left open. In this paper, we solve the open problem by showing that MSR is NP-complete, using a polynomial time reduction from One-in-Three 3SAT. We also solve the MSR problem and its variants exactly with FPT algorithms, i.e., showing that MSR is fixed-parameter tractable. Let k be the minimum number of markers deleted in various versions of MSR, the running time of our

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n + k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k = 0), we parameterize graph problems by k, and find FPT algorithms for several