Question: Why Is Longest Path NP Hard?

What are the shortest path algorithms?

Given a graph and a source vertex in the graph, find shortest paths from source to all vertices in the given graph.

Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum spanning tree..

Which algorithm is best for Shortest Path?

What Is the Best Shortest Path Algorithm?Dijkstra’s Algorithm. Dijkstra’s Algorithm stands out from the rest due to its ability to find the shortest path from one node to every other node within the same graph data structure. … Bellman-Ford Algorithm. … Floyd-Warshall Algorithm. … Johnson’s Algorithm. … Final Note.

What is the shortest path from node A to node F?

Answer : B If we use the graph on question 2 and increase all edge weights by 1, the shortest path from node A to node F is no longer A -> C -> E -> F, it becomes A -> F.

What is NP hard problem with example?

In computational complexity theory, NP-hardness (non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally “at least as hard as the hardest problems in NP”. A simple example of an NP-hard problem is the subset sum problem.

Is Max clique NP complete?

Theorem 20.2 Max-Clique is NP-Complete. We then put an edge between two nodes if the partial assignments are consistent. Notice that the maximum possible clique size is m because there are no edges between any two nodes that correspond to the same clause c.

Is Hamiltonian path NP complete?

The number of calls to the Hamiltonian path algorithm is equal to the number of edges in the original graph with the second reduction. Hence the NP-complete problem Hamiltonian cycle can be reduced to Hamiltonian path, so Hamiltonian path is itself NP-complete.

Is Path NP complete?

3 Answers. The shortest (in terms of weight) path, constrained to have exactly n (or at most n) edges, can be found in polynomial time. … If repetitions are disallowed, and G has n+1 vertices, then the shortest length-n path is just a Traveling salesman path, so of course it’s NP-complete.

Can Dijkstra find longest path?

Dijkstra cannot be used to compute the longest path. Or if it can, it wouldn’t be efficient. Simply do the edge relax to all edges in topological order. It will be the most efficient method.

Is shortest path NP complete?

Since it is also in NP, it is NP-Complete. The shortest path on the other hand is a different one, it asks what is the shortest way from point A to point B, and it is in P because there is a polynomial time algorithm that solves it (Dijkstra’s algorithm, Bellman-Ford, BFS for non weighted graphs).

Is clique NP hard?

In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. … Most versions of the clique problem are hard. The clique decision problem is NP-complete (one of Karp’s 21 NP-complete problems).

Is Dijkstra greedy?

In fact, Dijkstra’s Algorithm is a greedy algo- rithm, and the Floyd-Warshall algorithm, which finds shortest paths between all pairs of vertices (see Chapter 26), is a dynamic program- ming algorithm. Although the algorithm is popular in the OR/MS literature, it is generally regarded as a “computer science method”.

Can a shortest path contain a cycle?

All edges have positive weights: Shortest paths cannot contain cycles. All edges have non-negative weights. 0-cycles can occur in shortest paths but can be removed without changing the length.

Why is longest path NP complete?

Now it is easy to conclude that Longest Path is NP-complete because it is in NP and HamiltonianPath ∝ LongestP ath simply by observing that there is a Hamiltonian path in G if and only if there is a path of length n − 1.

Is Dijkstra NP hard?

P is the shortest cycle starting and ending at v which visits each v in V . … Therefore, P is also the solution to the travelling salesman problem. The reduction is obviously polynomial time (constant, actually), so your problem is NP-hard.

What does NP hard mean?

at least as hard as any NPA problem is NP-hard if an algorithm for solving it can be translated into one for solving any NP-problem (nondeterministic polynomial time) problem. NP-hard therefore means “at least as hard as any NP-problem,” although it might, in fact, be harder.

How do you find the shortest path?

Dijkstra’s algorithm can be used to find the shortest path. This algorithm will continue to run until all of the reachable vertices in a graph have been visited, which means that we could run Dijkstra’s algorithm, find the shortest path between any two reachable nodes, and then save the results somewhere.

Is NP complete the longest path?

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. …

What is Max clique?

This problem was the Maximal Clique Problem: given a group of vertices some of which have edges in between them, the maximal clique is the largest subset of vertices in which each point is directly connected to every other vertex in the subset.