usaco-guide/content/6_Plat/SPneg.md

id	title	author	prerequisites	description
sp-neg	Shortest Paths with Negative Edge Weights	Benjamin Qi	Gold - Shortest Paths with Non-Negative Edge Weights	Applications of Bellman-Ford.

• Hasn't appeared in recent USACO Gold as far as I know.
• If no negative cycles, can use Shortest Path Faster Algorithm or modify Dijkstra slightly (though the same running time bound no longer applies).

Tutorial

• CPH 13.1
• cp-algo Bellman Ford
• Topcoder Graphs Pt 3

Problems

• General
  • CSES High Score
  • Kattis SSSP Negative
  • CSES (Negative) Cycle Finding

Simple Linear Programming

You can also use shortest path algorithms to solve the following problem (a very simple linear program).

Given variables x_1,x_2,\ldots,x_N with constraints in the form x_i-x_j\ge c, compute a feasible solution.

• Linear Programming Trick

Problems

• Restore Array
• Art (basically same as above)
• Timeline (Camp)
  • equivalent to Timeline (Gold) except N,C\le 5000 and negative values of x are possible.