CHICAGO J OURNAL OF THEORETICAL COMPUTER SCIENCE 2014, Article 05, pages 1–18 http://cjtcs.cs.uchicago.edu/ Complexity of Testing Reachability in Matroids B.V. Raghavendra Rao ∗ Jayalal Sarma † Received January 22, 2012; Revised June 7, 2014; Published July 9, 2014 Abstract: We extend the complexity theoretic framework of reachability problems in graphs to the case of matroids. Given a matroid M and two elements s and t of the ground set, the reachability problem is to test if there is a circuit in M containing both s and t . We show complexity characterizations for several important classes of matroids. In particular, we show: (1) For two important classes of matroids associated with graphs, namely, graphic and bi-circular matroids we show that the reachability problem is L-complete. (2) For transversal matroids, when a basis of M is also given at the input, the problem can be shown to be NL- complete. A general upper bound for this case is the complexity of constructing a matching (RNC 2 ∩ P). (3) For linear matroids representable over Q and Z p , we show that the problem characterizes L C = L and Mod p L respectively, which provides the first characterizations of these classes in terms of reachability problems. 1 Introduction Reachability testing in various mathematical structures is a fundamental problem in complexity theory as they encode resource bounded computations in a natural way. The most useful one among them, especially in the context of space bounded computations, is the graph reachability problem, and has exact complexity characterizations both in the case of directed and undirected graphs. For directed graphs ∗ Department of Computer Science and Engineering, Indian Institute of Technology Madras , Chennai, 600 036, INDIA. bvrr@cse.iitm.ac.in † Department of Computer Science and Engineering, Indian Institute of Technology Madras, Chennai, 600 036, INDIA. jayalal@cse.iitm.ac.in Key words and phrases: Space Complexity, Reachability Problems, Matroid Theory © 2014 B. V. Raghavendra Rao and Jayalal Sarma cb Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.4086/cjtcs.2014.005