#pvnp

We see above the skill tree for the game Path of Exile.  The Problem that we are posed with is such. Given the Skill Tree above, what is the best path/nodes for maximum game play. 

Problem: The Skill Tree Problem

Input: Given a Skill Tree in the form of a graph G and k, the number of nodes you are allowed to select.

Output: Find the most OP Path.

This is rather simple for say a game like Borderlands 

Players are restricted by their level and have the option on building out three separate trees (although since they are all connected to a root node, it's basically one tree).

However, we see that this can scale up dramatically, as is the case of Path of Exile. In fact, the Skill Tree Problem is probably NP Complete since this can probably be reduced to the infamous Traveling Salesman Problem (Will attempt to prove this some other time or leave as an exercise to the reader =P)

How to Solve: NP complete problems are in the set of NP problems and the set of NP hard problems and are therefore difficult to solve.

In fact, NP stands for Non-deterministic Polynomial meaning this problem can be solved by a Non-deterministic Turing Machine in Polynomial Time. Usually, we approximate NP Complete problems through the use of dynamic programming and greedy algorithms. For practicality these approximations suffice.

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are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of November 2012, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute

P vs. NP problem The question is whether, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can alsofind that solution quickly. The former describes the class of problems termed NP, whilst the latter describes P. The question is whether or not all problems in NP are also in P. This is generally considered one of the most important open questions in mathematics and theoretical computer science

Hodge Conjecture The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

Riemann Hypothesis The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2.

Yang-Mills existence and mass gap In physics, classical Yang–Mills theory is a generalization of the Maxwell theory of electro-magnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.

Navier-Stokes existence and smoothness The Navier–Stokes equations describe the motion of fluids.