Approximate local search in combinatorial optimization

Local search algorithms for combinatorial optimization problems are generally of pseudopolynomial running time, and polynomial-time algorithms are not often known for finding locally optimal solutions for NP-hard optimization problems. We introduce the concept of Ε-local optimality and show that, for every Ε> 0, an Ε-local optimum can be identified in time polynomial in the problem size and 1/Ε whenever the corresponding neighborhood can be searched in polynomial time. If the neighborhood can be searched in polynomial time for a δ-local optimum, a variation of our main algorithm produces a (δ+Ε)-local optimum in time polynomial in the problem size and 1/Ε. As a consequence, a combinatorial optimization problem has a fully polynomial-time approximation scheme if and only if the problem of determining a better neighbor in an exact neighborhood has a fully polynomial-time approximation scheme.