An O(nm) time algorithm for finding the min length directed cycle in a graph

@inproceedings{Orlin2017AnOT,
  title={An O(nm) time algorithm for finding the min length directed cycle in a graph},
  author={James B. Orlin and Antonio Sede{\~n}o-Noda},
  booktitle={ACM-SIAM Symposium on Discrete Algorithms},
  year={2017}
}
In this paper, we introduce an O(nm) time algorithm to determine the minimum length directed cycle (also called the "minimum weight directed cycle") in a directed network with n nodes and m arcs and with no negative length directed cycles. This result improves upon the previous best time bound of O(nm + n2 log log n). Our algorithm first determines the cycle with minimum mean length λ* in O(nm) time. Subsequently, it chooses node potentials so that all reduced costs are λ* or greater. It then… 

Figures and Tables from this paper

Directed Shortest Paths via Approximate Cost Balancing

We present an O(nm) algorithm for all-pairs shortest paths computations in a directed graph with n nodes, m arcs, and nonnegative integer arc costs. This matches the complexity bound attained by

Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

This work generalizes the exact fully dynamic APSP data structure of Abraham et al. to solve the multiple-pairs shortest paths problem, where one is interested in computing distances for some k fixed source-target pairs after each update, and shows that in such a scenario, Õ((m+ k)n) worst-case update time is possible.

Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners

This paper gives the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous roundtrip spanner constructions, and shows that if the girth is 0(na), then the same guarantee can be achieved via a deterministic algorithm.

Constant girth approximation for directed graphs in subquadratic time

This is the first algorithm which approximates the girth of a directed graph up to a constant multiplicative factor faster than All-Pairs Shortest Paths (APSP) time, i.e. O(mn), and constitutes the first sub-APSP-time algorithm for approximatingThe girth to constant accuracy.

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Hardness for a variety of sparse graph problems from the hardness of a dense graph problem is proved.

Listing Acyclic Subgraphs and Subgraphs of Bounded Girth in Directed Graphs

This work proposes polynomial delay algorithms for listing both induced and edge subgraphs with girth g in time O(n) per solution; both improve upon a naive solution, respectively by a factor O(nm) and \(O(m^2)\).

Faster approximation algorithms for computing shortest cycles on weighted graphs

The approach combines some new insights on the previous approximation algorithms for this problem with Hitting Set based methods that are used for approximate distance oracles and date back from (Thorup and Zwick, JACM'05).

Fine-Grained Complexity and Conditional Hardness for Sparse Graphs

The notion of a sparse reduction which preserves the sparsity of graphs is introduced, and near linear-time sparse reductions between various pairs of graph problems in the $\tilde{O}(mn)$ class are presented.

Fine-grained complexity for sparse graphs

The notion of MWC hardness is formulated, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn, and the framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse.

On the Power of Tree-Depth for Fully Polynomial FPT Algorithms

It is shown that a simple divide-and-conquer method can solve many graph problems, including Weighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover, in time, where $\mathrm{td}$ is the tree-depth of the input graph.

References

SHOWING 1-10 OF 33 REFERENCES

Improved Shortest Paths on the Word RAM

It is shown that the component tree for an undirected network can be constructed in deterministic linear time and space with a simple algorithm, to be contrasted with a complicated and impractical solution suggested by Thorup.

New scaling algorithms for the assignment and minimum mean cycle problems

The assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm and can be solved in O( $$\sqrt n $$ m lognC) time, which is the best available time bound to solve the minimum mean cycle problem.

Minimum Weight Cycles and Triangles: Equivalences and Algorithms

  • L. RodittyV. V. Williams
  • Computer Science, Mathematics
    2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
  • 2011
The fundamental algorithmic problem of finding a cycle of minimum weight in a weighted graph is considered and efficient reductions imply the following surprising phenomenon: a minimum cycle with an arbitrary number of weighted edges can be ``encoded'' using only three edges within roughly the same weight interval.

On the K best integer network flows

The Scaling Network Simplex Algorithm

In this paper, we present a new primal simplex pivot rule and analyze the worst case complexity of the resulting simplex algorithm for the minimum cost flow, the assignment, and the shortest path

A note on two problems in connexion with graphs

  • E. Dijkstra
  • Mathematics, Computer Science
    Numerische Mathematik
  • 1959
A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.

Finding a minimum circuit in a graph

Finding minimum circuits in graphs and digraphs is discussed and an algorithm to find an almost minimum circuit is presented and an alternative method is to reduce the problem of finding a minimum circuit to that of finding an auxiliary graph.

An Experimental Study of Minimum Mean Cycle Algorithms

Howard's algorithm, a well known algorithm to the stochastic control community but a relatively unknown algorithms to the graph theory community, is by far the fastest algorithm on the test suite although the only known bound on its running time is exponential.

Depth-First Search and Linear Graph Algorithms

The value of depth-first search or “backtracking” as a technique for solving problems is illustrated by two examples. An improved version of an algorithm for finding the strongly connected components

A characterization of the minimum cycle mean in a digraph

  • R. Karp
  • Mathematics
    Discret. Math.
  • 1978