Visualizing SLS Runtime Behaviour

Stochastic Local Searches (SLS) are a class of meta-heuristics for solving hard combinatorial optimization problems. Common examples of such problems are satisfiability (SAT), travelling salesman problem (TSP), job shop problem (JSP) and vehicle routing problem (VRP). The set of all possible solutions to one of these problems is called the search space. Each point (solution) in the search space is associated with some measure of fitness or objective. The idea behind local search is that some initial point in the search space is chosen as a starting point. From this point, and all future ones, we define a neighbourhood. A neighbourhood for a point is the set of all solutions which can be constructed by modifying the current solution in some well defined way. This modification is typically very small; such as swapping a single assignment for SAT. A point from the neighbourhood is chosen to become the new solution. This process is repeated for some number of iterations until either a solution of sufficiently good quality is located, a time bound is met, or some measure of stagnation is reached. The key parts of an SLS algorithm are then: (1) how to choose the initial position, (2) how to define the neighbourhood for a given point, (3) how to decide which point within a neighbourhood to choose, (4) when to decide that the search should terminate. For a concrete example we can construct a very simple SAT solver: (1) an initial solution is chosen by generating a random assignment for each variable, (2) the neighbourhood for each point contains every solution that can be obtained by changing a single variable assignment, (3) the neighbour with the fewest violated clauses is chosen, (4) the search terminates when no neighbours can be found with fewer violated clauses than the current solution. This is an example of a simple gradient decent (or hill climbing) search. This simple solver for SAT should help to illustrate a key point about SLS. There is no guarantee that a SLS method will find an optimal solution. There isn’t even a guarantee that an SLS method can find a feasible solution! This means that SLS algorithms are incomplete. So why use them? Well, they’re fast. Very fast. And in practice a ”good” solution found today is often more desirable than a perfect solution found next year.