Maximum weight clique algorithms for circular-arc graphs and circle graphs
SIAM Journal on Computing
The max clique problem in classes of string-graphs
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
Maximum weight independent sets and cliques in intersection graphs of filaments
Information Processing Letters
On the Approximation Properties of Independent Set Problem in Degree 3 Graphs
WADS '95 Proceedings of the 4th International Workshop on Algorithms and Data Structures
Dotted interval graphs and high throughput genotyping
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
SIAM Journal on Computing
Linear degree extractors and the inapproximability of max clique and chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The Complexity of Combinatorial Optimization Problems on $d$-Dimensional Boxes
SIAM Journal on Discrete Mathematics
Optimization problems in multiple-interval graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for intersection graphs
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Universality considerations in VLSI circuits
IEEE Transactions on Computers
Approximating the 2-interval pattern problem
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms
Discrete Optimization
The clique problem in ray intersection graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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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 WEIGHTED MAXIMUM CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t.