Approximate local search in combinatorial optimization
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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 $\varepsilon$-local optimality and show that, for every $\varepsilon 0$, an $\varepsilon$-local optimum can be identified in time polynomial in the problem size and $1/\varepsilon$ whenever the corresponding neighborhood can be searched in polynomial time. If the neighborhood can be searched in polynomial time for a $\delta$-local optimum, a variation of our main algorithm produces a $(\delta + \varepsilon)$-local optimum in time polynomial in the problem size and $1/\varepsilon$. 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.