What are negative weight cycles in graphs?

What are negative weight cycles in graphs?

Negative weight cycles are cycles in a graph where the sum of the edge weights is negative, which can lead to infinite reductions in path costs.

Can you give me an example of a negative weight cycle?

Sure! Consider a graph with three vertices A, B, and C. If the edges are A to B with weight 1, B to C with weight -2, and C to A with weight 1, the cycle A -> B -> C -> A has a total weight of 0, but if we adjust the weights slightly, we can create a negative cycle.

How do negative weight cycles affect algorithms like Bellman-Ford?

The Bellman-Ford algorithm can detect negative weight cycles. If it finds one, it indicates that the shortest path cannot be determined because you can keep reducing the path cost indefinitely.

Are there any real-world applications of negative weight cycles?

Yes, negative weight cycles can model situations like financial arbitrage, where you can exploit price differences to make a profit without risk.