# Open problem for February 2014: Better approximations for the distance to monotonicity

Slightly delayed, Feb’s open problem is by one of the three possible “yours truly”s, Sesh. Looking for more reader participation, hint, hint.

Basic setting: Consider $$f:\{0,1\}^n \rightarrow R$$, where $$R$$ is some ordered range. There is a natural coordinate-wise partial order denoted by $$\prec$$. The function is monotone if for all $$x \prec y$$, $$f(x) \leq f(y)$$. The distance to monotonicity, $$\epsilon_f$$ is the minimum fraction of values that must be changed to make $$f$$ monotone. This is an old problem in property testing.

Open problem: Is there an efficient constant factor approximation algorithm for $$\epsilon_f$$? In other words, is there a $$poly(n/\epsilon_f)$$ time procedure that can output a value $$\epsilon’ = \Theta(\epsilon_f)$$?

State of the art: Existing monotonicity testers give an $$O(n)$$-approximation for $$\epsilon_f$$, so there is much much room for improvement. (I’d be happy to see a $$O(\log n)$$-approximation.) Basically, it is known that the number of edges of $$\{0,1\}^n$$ that violate monotonicity is at least $$\epsilon_f 2^{n-1}$$ [GGLRS00], [CS13]. A simple exercise (given below) shows that there are at most $$n\epsilon_f 2^n$$ violated edges. So just estimate the number of violated edges for an $$O(n)$$-approximation.
(Consider $$S \subseteq \{0,1\}^n$$ such that modifying $$f$$ on $$S$$ makes it monotone. Every violated edge must have an endpoint in $$S$$.)

Related work: Fattal and Ron [FR10] is a great place to look at various related problems, especially for hypergrid domains.

References

[CS13] D. Chakrabarty and C. Seshadhri. Optimal Bounds for Monotonicity and Lipschitz Testing over Hypercubes and Hypergrids. Symposium on Theory of Computing, 2013.

[FR10] S. Fattal and D. Ron. Approximating the distance to monotonicity in high dimensions . ACM Transactions on Algorithms, 2010.

[GGL+00] O. Goldreich, S. Goldwasser, E. Lehman, D. Ron, and A. Samorodnitsky. Testing Monotonicity . Combinatorica, 2000.