Black-box Methods for Restoring Monotonicity
This addresses the issue of ensuring monotonicity in practical applications where non-monotone solutions arise, but it is incremental as it builds on existing black-box and monotonicity restoration concepts.
The paper tackles the problem of restoring monotonicity in heuristic or approximation algorithms that often fail to satisfy this property, by developing black-box methods that degrade function values by at most ε with query complexities logarithmic in 1/ε and exponential in parameters, and also improves bounds for k-marginal monotonicity.
In many practical applications, heuristic or approximation algorithms are used to efficiently solve the task at hand. However their solutions frequently do not satisfy natural monotonicity properties of optimal solutions. In this work we develop algorithms that are able to restore monotonicity in the parameters of interest. Specifically, given oracle access to a (possibly non-monotone) multi-dimensional real-valued function $f$, we provide an algorithm that restores monotonicity while degrading the expected value of the function by at most $\varepsilon$. The number of queries required is at most logarithmic in $1/\varepsilon$ and exponential in the number of parameters. We also give a lower bound showing that this exponential dependence is necessary. Finally, we obtain improved query complexity bounds for restoring the weaker property of $k$-marginal monotonicity. Under this property, every $k$-dimensional projection of the function $f$ is required to be monotone. The query complexity we obtain only scales exponentially with $k$.