A Probabilistic Approach to Problems Parameterized Above or Below Tight Bounds

We introduce a new approach for establishing fixed-parameter tractability of problems parameterized above tight lower bounds. To illustrate the approach we consider three problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar (J. Comput. Syst. Sci. 75, 2009). We show that a generalization of one of the problems and non-trivial special cases of the other two are fixed-parameter tractable.

💡 Research Summary

The paper tackles a class of parameterized problems that are defined relative to a known tight bound – either “above” a lower bound or “below” an upper bound – a setting that has resisted traditional fixed‑parameter tractability (FPT) techniques. The authors introduce a probabilistic framework that first shows, via expectation arguments, that a random assignment is likely to meet the required bound, and then derandomizes this existence proof to obtain deterministic FPT algorithms.

The motivation comes from three open problems posed by Mahajan, Raman, and Sikdar (2009): (1) MAX‑k‑SAT parameterized above a tight lower bound, (2) MAX‑CUT parameterized below a tight upper bound, and (3) Vertex Cover parameterized above a tight lower bound. For each problem the optimal value is known to be bounded by a simple combinatorial expression (e.g., the number of clauses, half the edges, or the size of a minimum vertex cover). The parameter k measures how far a solution exceeds (or falls short of) that bound. Prior work could not place these problems in any known FPT class; they were either unresolved or suspected to be W