Scheduling jobs of equal length: complexity, facets and computational results
Mathematical Programming: Series A and B
An Optimal Algorithm to Detect a Line Graph and Output Its Root Graph
Journal of the ACM (JACM)
Approximation Algorithms for the Job Interval Selection Problem and Related Scheduling Problems
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
SIAM Journal on Computing
On restrictions of balanced 2-interval graphs
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
On the parameterized complexity of some optimization problems related to multiple-interval graphs
Theoretical Computer Science
Recognizing d-interval graphs and d-track interval graphs
FAW'10 Proceedings of the 4th international conference on Frontiers in algorithmics
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We consider the class of strip graphs, a generalization of interval graphs. Intervals are assigned to rows such that two vertices have an edge between them if either their intervals intersect or they belong to the same row. We show that recognition of the class of strip graphs is -complete even if all intervals are of length 2. Strip graphs are important to the study of job selection, where we need an equivalence relation to connect multiple intervals that belong to the same job. The problem we consider is Job Interval Selection (JISP) on m machines. In the single-machine case, this is equivalent to Maximum Independent Set on strip graphs. For m machines, the problem is to choose a maximum number of intervals, one from each job, such that the resulting choices form an m-colorable interval graph. We show the single-machine case to be fixed-parameter tractable in terms of the maximum number of overlapping rows. We also use a concatenation operation on strip graphs to reduce the m-machine case to the 1-machine case. This shows that m-machine JISP is fixed-parameter tractable in the total number of jobs.