A probabilistic approach to problems parameterized above or below tight bounds

Abstract

We introduce a new approach for establishing fixed-parameter tractability of problems parameterized above tight lower bounds or below tight upper bounds. To illustrate the approach we consider two problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar [M. Mahajan, V. Raman, S. Sikdar, Parameterizing above or below guaranteed values, J. Comput. System Sci. 75 (2) (2009) 137-153]. We show that a generalization of one of the problems and three non-trivial special cases of the other problem admit kernels of quadratic size. As a byproduct we obtain a new probabilistic inequality that could be of independent interest. Our new inequality is dual to the Hypercontractive Inequality.

Every ternary permutation constraint satisfaction problem parameterized above average has a kernel with a quadratic number of variables

“…because each arc contributes exactly one to u∈V l(u) and one to u∈V r(u). We conclude that E[X 2 ] ≥ 3 16 m − 10 64 m = 1 32 m. The following theorem was proved in [21]. 7 Kernels for Π-AA Problems…”

“…We build on the probabilistic Strictly Above Expectation method by Gutin et al [21] to prove non-trivial lower bounds on the minimum fraction of satisfiable constraints in instances belonging to a restricted subclass. For such an instance with parameter k, we introduce a random variable X such that the instance is a "yes"-instance if and only if X takes with positive probability a value greater than or equal to k. If X happens to be a symmetric random variable with finite second moment then P(X ≥ E[X 2 ]) > 0; it hence suffices to prove E[X 2 ] = h(k) for some monotonically increasing unbounded function h. (Here, P(•) and E[•] denote probability and expectation, respectively.)…”

“…This implies the existence of kernels with a quadratic number of variables for most ternary Permutation-CSPs. The remaining ternary Permutation-CSPs are proved to be equivalent to Acyclic Subdigraph-AA (a binary Permutation-CSP defined in Section 6) and since Acyclic Subdigraph-AA, as shown in [21], has a kernel with a quadratic number of variables, the remaining ternary Permutation-CSPs have a kernel with a quadratic number of variables.…”

“…Massive interest in parameterizations above tight lower bounds came with the paper of Mahajan et al [28], who stated several questions on fixed-parameter tractability of maximization problems parameterized above tight lower bounds, some of which are still open. Several of those questions were answered by newly-developed methods [1,9,10,20,21], using algebraic, probabilistic and harmonic analysis tools. In particular, a probabilistic approach allowed Gutin et al [20] to prove the existence of a quadratic kernel for the parameterized Betweenness Above Average (Betweenness-AA) problem, thus, answering an open question of Benny Chor [29].…”

Constraint Satisfaction Problems Parameterized above or below Tight Bounds: A Survey

“…In Section 3, we describe some probabilistic and Harmonic Analysis tools. These tools are, in particular, used in the recently introduced Strictly-Above-Below-Expectation method [30].…”