Breaking the O(m 2 n) Barrier for Minimum Cycle Bases

Abstract

We give improved algorithms for constructing minimum directed and undirected cycle bases in graphs. For general graphs, the new algorithms are Monte Carlo and have running time O(m ω ), where ω is the exponent of matrix multiplication. The previous best algorithm had running timeÕ(m 2 n). For planar graphs, the new algorithm is deterministic and has running time O(n 2 ). The previous best algorithm had running time O(n 2 log n). A key ingredient to our improved running times is the insight that the search for minimum bases can be restricted to a set of candidate cycles of total length O(nm).

Certifying algorithms

“…To certify condition (c), we assume for simplicity from now on that the shortest path between every two vertices of the input graph is unique; such a situation can be simulated by adding a random infinitesimally small weight to every edge. For further details see [108]. ([108]).…”

“…The fastest known algorithm for computing a minimum weight basis is a Monte Carlo algorithm with running time O(m ω ), where ω is the exponent of matrix multiplication [108]. The algorithm can be made certifying and the witness can be checked in Monte Carlo time O(m 2 ).…”

Applications of Ear Decomposition to Efficient Heterogeneous Algorithms for Shortest Path/Cycle Problems

“…Mehlhorn et al [29] further describes a O(m 2 n/ log n + n 2 m) algorithm for undirected weighted graphs and also provided for a simpler way to obtain the shortest cycle in each phase. Amaldi et al [1] characterizes the Horton cycles to obtain a restricted set of cycles known as the isometric cycles and provides an improved O(m ω ) Monte Carlo algorithm where ω is the exponent of the fast matrix multiplication.…”

“…We will now summarize the deterministic sequential algorithms from [1,11,18,29] for obtaining an MCB. Horton [18] provided the first polynomial time algorithm for computing an MCB.…”

“…Notice that there are n · (m − n + 1) cycles in Horton-Set(G). Many recent works [1,29] use the idea that the Horton cycles of G with respect to a feedback vertex set of V (G) suffices. A Feedback Vertex Set (FVS) is a set of vertices such that every cycle in the graph contains some vertex in the set [20].…”