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.

News for January 2014

As we already covered in earlier posts, quite a few notable property testing papers were presented at the SODA and ITCS conferences this month. In addition, an exciting new set of results on a variant of the linearity testing problem was also posted on ECCC.

Direct Sum Testing by Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. One of the true gems of property testing is the linearity testing: as Blum, Luby, and Rubinfeld first showed, we can test whether a boolean function is linear with a (very natural) 3-query test. A fascinating variant of this problem that is still largely open to this day is the following: is linearity still testable even when we consider functions defined on a subset of the hypercube? This paper considers the problem when the subsets in question are layers of the hypercube, a problem that is closely connected to PCP constructions and direct sum operations on multi-player games. The main result of the paper shows that we can also test linearity for functions defined on layers of the hypercube with a natural 3-query tester.