Everything about Longest path problem
Problem of finding the longest simple path for a given graph
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Longest path problemA 5-minute orientation
- Skim a recent paper: New formulations and branch-and-cut procedures for the longest induced path problemInfluential work since 2022
- Pick up a classic book: Witness to HopeThe most-reprinted book on the subject
- Read the landmark paper: The Longest Path Problem has a Polynomial Solution on Interval GraphsCited 30 times
- Go sideways: Independent set (graph theory)A closely related topic
- Go sideways: Directed acyclic graphA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
The Longest Path Problem has a Polynomial Solution on Interval Graphs
Linear structure of bipartite permutation graphs and the longest path problem
A linear-time algorithm for the longest path problem in rectangular grid graphs
A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs
The probabilistic longest path problem
The Orderly Colored Longest Path Problem – a survey of applications and new algorithms
An efficient parallel algorithm for the longest path problem in meshes
The Longest Path Problem Is Polynomial on Cocomparability Graphs
🆕 Influential since 2022
New formulations and branch-and-cut procedures for the longest induced path problem
The Longest (s, t)-Path Problem on O-Shaped Supergrid Graphs
A Three-Stage MCDM and Extended Longest Path Algorithm for the Satellite Image Acquisition Scheduling Problem
An algebraic approach to the longest path problem
Generalized Longest Path Problems
📚 Books worth reading
Witness to HopeGeorge Weigel · 1999
Language and Automata Theory and ApplicationsAdrian-Horia Dediu · 2012
Algorithm Design and ApplicationsMichael T. Goodrich, Roberto Tamassia · 2014
Introduction to Programming in JavaRobert Sedgewick, Kevin Wayne · 2007
Path problems in networksJohn S. Baras · 2010
প্রোগ্রামিং কনটেস্টমোঃ মাহবুবুল হাসান · 2016
BIOINFORMATICS ALGORITHMS,VOL.IIPhillip Compeau, Pavel Pevzner · 2015
AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMSMichael Soltys · 2012
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Independent set (graph theory) In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a…
- Directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it…
- Feedback arc set In graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at…
- Clique cover In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. Generally…
- Distance-hereditary graph In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances…
- Pathwidth In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number…
- Topological sorting In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge…
- Matroid parity problem In combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid, a structure…