Steepest ascent can be exponential in bounded treewidth problems

Cohen, David A.; Cooper, Martin; Kaznatcheev, Artem; Wallace, Mark

doi:10.1016/j.orl.2020.02.010

Cited by 2 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If the objective values are randomly generated the number of local optima tends to grow exponentially with n. Thus the expected number of successful flips to reach a local optimum from an arbitrary point grows only linearly with n. A more general analysis of neighbourhood search in [Tov03] explores a variety of algorithms to reach a local optimum, but does not compare neighbourhood search against random generate-and-test. In Tovey's scenario, with only zero-one variables where the neighbourhood is defined by single flips, there are problems that require an exponential number of flips to reach a local optimum -even choosing the best neighbour each time [CCKW20]. [Gro92] analysed the average difference between a candidate solution and its neighbours, for five well-known combinatorial optimisation problems.…”

Section: Related Workmentioning

confidence: 99%

“…We seek to prove that, from a point with cost k, the expected number of steps to reach an optimum point steps(k) is lower than with blind search. [CCKW20] exhibited a problem with binary variables with a simple objective, that requires an exponential number of flips to reach a local optimum -even choosing the best neighbour each time. Nevertheless we prove that if local descent continues to improve long enough, its expected number of steps -although only choosing the first improving neighbour -will be less than with blind search.…”

Section: Theorem On the Benefit Of Local Descentmentioning

confidence: 99%

See 1 more Smart Citation

Minimal Conditions for Beneficial Local Search

Wallace¹

2021

Preprint

View full text Add to dashboard Cite

This paper investigates why it is beneficial, when solving a problem, to search in the neighbourhood of a current solution. There is a huge range of algorithms that use this for local improvement, generally within a larger meta-heuristic framework, for applications in machine learning and optimisation in general. The paper identifies properties of problems and neighbourhoods that support two novel proofs that neighbourhood search is beneficial over blind search. These are: firstly a proof that search within the neighbourhood is more likely to find an improving solution in a single search step than blind search; and secondly a proof that a local improvement, using a sequence of neighbourhood search steps, is likely to achieve a greater improvement than a sequence of blind search steps. To explore the practical impact of these properties, a range of problem sets and neighbourhoods are generated, where these properties are satisfied to different degrees. Experiments reveal that the benefits of neighbourhood search vary dramatically in consequence. Random sets of problems of a classical combinatorial optimisation problem are analysed, in order to demonstrate that the underlying theory is reflected in practice.

show abstract

Section: Related Workmentioning

confidence: 99%

Section: Theorem On the Benefit Of Local Descentmentioning

confidence: 99%

Minimal Conditions for Beneficial Local Search

Wallace¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Note that although the fitness graphs corresponding to Example 7.1 have long improving paths, standard local search algorithms would be unlikely to follow these paths. However, with careful padding, Example 7.1 has been converted to a family of Boolean VCSP instances of treewidth 7 where even a popular local search algorithm like steepest ascent will follow an exponentially long improving path (Cohen, Cooper, Kaznatcheev, & Wallace, 2020).…”

mentioning

confidence: 99%

Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

Kaznatcheev

Cohen

Jeavons³

2020

jair

View full text Add to dashboard Cite

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we consider how fitness landscapes can be represented using valued constraints, and investigate what the structure of such representations reveals about the complexity of local search. First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal constraint graph still exists, but is NP-hard to compute. We then develop several techniques to bound the length of any sequence of local search moves. We show that such a bound can be obtained from the numerical values of the constraints in the representation, and show how this bound may be tightened by considering equivalent representations. In the binary Boolean case, we prove that a degree 2 or treestructured constraint graph gives a quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for any form of local search. Finally, we build two families of examples to show that the conditions in our tractability results are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth-two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves.

show abstract

Steepest ascent can be exponential in bounded treewidth problems

Cited by 2 publications

References 10 publications

Minimal Conditions for Beneficial Local Search

Minimal Conditions for Beneficial Local Search

Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

Contact Info

Product

Resources

About