Multiple Interval Graphs Research Articles

The Maximum Clique Problem in Multiple Interval Graphs

Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t≥3 and polynomial-time solvable when t=1. The problem is also known to be NP-complete in t-track graphs when t≥4 and polynomial-time solvable when t≤2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for MAXIMUM WEIGHTED CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t.

Parameterized complexity in multiple-interval graphs: Domination, partition, separation, irredundancy

We show that the problem k-dominating set and its several variants including k-connected dominating set, k-independent dominating set, and k-dominating clique, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multiple-interval graphs and their complements. This answers an open question of Fellows et al. In sharp contrast, we show that d-distance k-dominating set for d≥2 is W[2]-complete in multiple-interval graphs and their complements. We also show that k-perfect code and d-distance k-perfect code for d≥2 are W[1]-complete even in unit 2-track interval graphs. In addition, we present various new results on the parameterized complexities of k-vertex clique partition and k-separating vertices in multiple-interval graphs and their complements, and present a very simple alternative proof of the W[1]-hardness of k-irredundant set in general graphs.

On the parameterized complexity of some optimization problems related to multiple-interval graphs

We show that for any constant t ≥ 2 , k - Independent Set and k - Dominating Set in t -track interval graphs are W[1]-hard. This settles an open question recently raised by Fellows, Hermelin, Rosamond, and Vialette. We also give an FPT algorithm for k - Clique in t -interval graphs, parameterized by both k and t , with running time max { t O ( k ) , 2 O ( k log k ) } ⋅ poly ( n ) , where n is the number of vertices in the graph. This slightly improves the previous FPT algorithm by Fellows, Hermelin, Rosamond, and Vialette. Finally, we use the W[1]-hardness of k - Independent Set in t -track interval graphs to obtain the first parameterized intractability result for a recent bioinformatics problem called Maximal Strip Recovery (MSR). We show that MSR- d is W[1]-hard for any constant d ≥ 4 when the parameter is either the total length of the strips, or the total number of adjacencies in the strips, or the number of strips in the solution.

Optimization problems in multiple-interval graphs

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multiple-interval graph. For Minimum Vertex Cover, we give a (2−1/ t )-approximation algorithm which also works when a t -interval representation of our given graph is absent. Following this, we give a t 2 -approximation algorithm for Minimum Dominating Set which adapts well to more general variants of the problem. We then proceed to prove that Maximum Clique is NP -hard already for 3-interval graphs, and provide a ( t 2 − t +1)/2-approximation algorithm for general values of t ≥ 2, using bounds proven for the so-called transversal number of t -interval families.

On the parameterized complexity of multiple-interval graph problems

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.

Covering and coloring problems for relatives of intervals

The following generalizations and relatives of interval families are studied in the paper: arcs of a circle, multiple intervals, chords of a circle, d-dimensional boxes and multiple boxes. For these families we survey results and problems concerning the dependence of the transversal number on the packing number and the dependence of the coloring number on the clique number. The intersection graphs of the underlying set systems are circular arc graphs, multiple interval graphs, circle graphs (called also overlap graphs), box graphs and multiple box graphs. Thus most of our problems and results concern the relation between the clique-cover number and stability number (ϑ and α), or between the chromatic number and clique number (χ and ω) of these graphs.

On the chromatic number of multiple interval graphs and overlap graphs

Let χ( G) and ω( G) denote the chromatic number and clique number of a graph G. We prove that χ can be bounded by a function of ω for two well-known relatives of interval graphs. Multiple interval graphs (the intersection graphs of sets which can be written as the union of t closed intervals of a line) satisfy χ⩽2 t( ω−1) for ω⩾2. Overlap graphs satisfy χ⩽2 ω ω 2( ω−1).

On double and multiple interval graphs

AbstractIn this paper we discuss a generalization of the familiar concept of an interval graph that arises naturally in scheduling and allocation problems. We define the interval number of a graph G to be the smallest positive integer t for which there exists a function f which assigns to each vertex u of G a subset f(u) of the real line so that f(u) is the union of t closed intervals of the real line, and distinct vertices u and v in G are adjacent if and only if f(u) and f(v)meet. We show that (1) the interval number of a tree is at most two, and (2) the complete bipartite graph Km, n has interval number ⌈(mn + 1)/(m + n)⌉.