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What is the relationship between $\mathsf{PLS}$ and $\mathsf{APX}$? In other words, are problems that admit a polynomial time local search approximable? Do approximable optimization problems imply a local search algorithm in general?

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If one wants to approximate the potential function, then yes, there even exists a fully polynomial-time approximation scheme (FPTAS). See

James B. Orlin, Abraham P. Punnen, Andreas S. Schulz: Approximate Local Search in Combinatorial Optimization. SIAM J. Comput. 33(5): 1201-1214 (2004).

For some settings though, this is not interesting. For example, for congestion games, where pure equilibria exist and their computation is PLS-complete, strategy profiles that approximate the potential function well may be very poor approximate equilibria. For some settings, constant-factor approximate equilibria can be computed in polynomial-time even when computing an exact equilibrium is PLS-hard, for other settings it is PLS-hard to compute an approximate equilibrium for any polynomial-time computable non-trivial approximation factor, as explained by the following announcement paper.

Ioannis Caragiannis, Angelo Fanelli, Nick Gravin, Alexander Skopalik: Computing approximate pure Nash equilibria in congestion games. SIGecom Exchanges 11(1): 26-29 (2012).

Note PLS might be much easier than FNP.

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