Solutions to genome scaffolding problems can be represented as paths and cycles in a "solution graph". However, when working with repetitions, such solution graphs may contain branchings and, thus, they may not be uniquely convertible into sequences. Having introduced various ways of extracting the unique parts of such solutions, we extend previously known NP-hardness results to the case that the solution graph is planar, bipartite, and subcubic, and show that there is no PTAS in this case.

Solutions to genome scaffolding problems can be represented as paths and cycles in a "solution graph". However, when working with repetitions, such solution graph may contain branchings and they may not be uniquely convertible into sequences. Having introduced, in a previous work, various ways of extracting the unique parts of such solutions, we extend previously known NP-hardness results to the case that the solution graph is planar, bipartite, and subcubic, and show the APXcompleteness in this case. We also provide some practical tests.

In our modern societies, a certain number of people do not own a car, by choice or by obligation. For some trips, there is no or few alternatives to the car. One way to make these trips possible for these people is to be transported by others who have already planned their trips. We propose to model this problem using as path-finding problem in a list edge-colored graph. This problem is a generalization of the [Formula: see text]-path problem, studied by Böhmová et al. We consider two optimization functions: minimizing the number of color changes and minimizing the number of colors. We study for the previous problems, the classic complexity (polynomial-case, NP-completeness, hardness of approximation) and parameter complexity (W[2]-hardness) even in restricted cases. We also propose a lower bound for exact algorithm. On the positive side we provide a polynomial-time approximation algorithm and a FPT algorithm.

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