# Solving the Longest Oneway-Ticket Problem and Enumerating Letter Graphs by Augmenting the Two Representative Approaches with ZDDs

## Abstract

Several researchers have studied subgraph enumeration algorithms that use a compressed expression for a family of sets, called a zero-suppressed binary decision diagram (ZDD), to solve subgraph optimization problems. We have two representative approaches to manipulate ZDDs effectively. One is fundamental mathematical operations on families of sets over ZDDs. The other is a direct construction method of a ZDD that represents desired subgraphs of a graph and is called frontier-based search. In this research, we augment the approaches by proposing two new operations, called disjoint join and joint join, on family algebra over ZDDs and extending the frontier-based search to enumerate subgraphs that have a given number of vertices of specified degrees. Employing the new approaches, we present enumeration algorithms for alphabet letter graphs on a given graph. Moreover, we solve a variant of the longest path problem, called the Longest Oneway-ticket Problem (LOP), that requires computing the longest trip on the railway network of the Japan Railways Group using a oneway ticket. Numerical experiments show that our algorithm solves the LOP and is faster than the existing integer programming approach for some instances.

## Keywords

Railway Network Enumeration Algorithm Degree Function Cycle Graph Integer Programming Approach## References

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