Ph.D Thesis | |

Ph.D Student | Yovel Uri |
---|---|

Subject | Worst Case Analysis of Local Search Based Heuristics |

Department | Department of Industrial Engineering and Management |

Supervisor | ASSOCIATE PROF. Asaf Levin |

Full Thesis text |

The method of
*local search* is widely used in many hard combinatorial optimization
problems. The idea is simple: start with an arbitrary (feasible) solution, and
at each step, search a (relatively small) neighborhood for an improved solution.
If such a solution is found, replace the current solution with it. Repeat this
procedure until the neighborhood (of the current solution) contains no
improving solutions. At this point, return the current solution, which is *locally
optimal*, and terminate.

Local search
algorithms are mainly used in the framework of *metaheuristics*, such as
simulated annealing, tabu search, genetic algorithms, etc. From a practical
point of view, they are usually very efficient and achieve excellent results - the
generated solutions are near optimal.

However, from
a theoretical point of view, there is usually no guarantee on their worst-case
performance. In this dissertation, we consider the metric of *approximation
ratios* as the tool for characterizing worst-case performance guarantees. Specifically,
for a minimization (maximization) problem, *A* is a *r-approximation
algorithm* if its running time is polynomial in the input size, and for any
instance*, apx ≤ r ? opt* ( *apx ≥ r ? opt* ), where
*apx* is the cost of the solution produced by *A* and *opt* is
the cost of an optimal solution.

The paradigm
of *non-oblivious* local search is a modification of the standard, i.e., *oblivious*,
local search scheme in the following sense: while the objective function of the
oblivious local search is simply the given problem's, a non-oblivious local
search is allowed to use a different objective function to guide the search. For
some problems, this method achieves better approximation ratios than the ones
obtained using oblivious local search.

In this
dissertation, we further demonstrate the power of non-oblivious local search by
providing such algorithms for three problems: *(p,k*)-Uniform Unweighted
Set Cover, the Traveling Salesman Problem, and the Vehicle Routing Problem.