When is local search both effective and efficient?

Combinatorial optimization problems define fitness landscapes that combine the numerics of the 'fitness' function to be maximized with the combinatorics of which assignments are adjacent. Local search starts at an initial assignment in this landscape and successively moves to assignments until no further improvement is possible among the adjacent assignments. Classic analyses of local search algorithms have focused mostly on the question of effectiveness ("did the algorithm find a good solution?") and often implicitly assumed that there are no doubts about their efficiency ("did the algorithm find the solution quickly?"). But there are many reasons to doubt the efficiency of local search. Many local search algorithms are known to be inefficient even if we focus on fitness landscapes on the hypercube that are single peaked on every subcube (known as semismooth fitness landscapes, completely unimodal pseudo-Boolean functions, or acyclic unique sink orientations). Here, we want to identify the most expressive subclass of single-peaked binary Boolean valued constraint satisfaction problems for which many popular local search algorithms are efficient. In this paper, we introduce the class of conditionally-smooth fitness landscapes where the preferred assignment of a variable xj depends only on the assignments of variables xi with i less than j in an associated partial order. We prove that many popular local search algorithms like random ascent, simulated annealing, various jumping rules, and the Kernighan-Lin heuristic are very efficient on conditionally-smooth landscapes. Some other popular local search algorithms like steepest ascent and random facet, however, still require a super-polynomial number of steps on these landscapes. Our hope is to contribute to a fuller understanding of what properties fitness landscapes must have for local search algorithms to be both effective and efficient.
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