PLS (complexity)

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In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem.

A PLS problem L has a set D_L of instances which are encoded using strings over a finite alphabet \Sigma. For each instance x there exists a finite solution set F_L(x). Each solution s \in F_L(x) has a non negative integer cost given by a function c_L(s, x) and a neighborhood N(s, x) \subseteq F_L(x). Additionally, the existence of the following three polynomial time algorithms is required:

  • Algorithm A_L produces some solution A_L(x) \in F_L(x).
  • Algorithm B_L determines the value of c_L(s, x).
  • Algorithm C_L maps a solution s \in F_L(x) to an element s' \in N(s, x) such that c_L(s', x) < c_L(s, x) if such an element exists. Otherwise C_L reports that no such element exists.

An instance D_L has the structure of an implicit graph, the vertices being the solutions with two solutions s, s' \in F_L(x) connected by a directed arc iff s' \in N(s, x). The most interesting computational problem is the following:

Given some instance x of a PLS problem L, find a local optimum of c_L(s, x), i.e. a solution s \in F_L(x) such that c_L(s', x) \geq c_L(s, x) for all s' \in N(s, x)

The problem can be solved using the following algorithm:

  1. Use A_L to find an initial solution s
  2. Use algorithm C_L to find a better solution s' \in N(s, x). If such a solution exists, replace s by s', else return s

Unfortunately, it generally takes an exponential number of improvement steps to find a local optimum even if the problem L can be solved exactly in polynomial time.

Examples of PLS-complete problems include local-optimum relatives of the travelling salesman problem, maximum cut and satisfiability, as well as finding a pure Nash equilibrium in a congestion game.

PLS is a subclass of TFNP, a complexity class closely related to NP that describes computational problems in which a solution is guaranteed to exist and can be recognized in polynomial time. For a problem in PLS, a solution is guaranteed to exist because the minimum-cost vertex of the entire graph is a valid solution, and the validity of a solution can be checked by computing its neighbors and comparing the costs of each one.