Approximate Local Search in Combinatorial Optimization
James B. Orlin, Abraham P. Punnen, Andreas S. Schulz · 2003
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.