AND-compression of NP-complete problems: Streamlined proof and minor observations

TL;DR

This paper provides a streamlined, generalized proof of Drucker's theorem on the non-existence of AND-compression for SAT, extending previous results to randomized algorithms and simplifying the original proof techniques.

Contribution

It offers a simplified, more general proof of Drucker's theorem, extending it to randomized algorithms and avoiding complex information-theoretic arguments.

Findings

No polynomial-time AND-compression for SAT unless coNP in NP/poly

Extension of Drucker's result to randomized algorithms with bounded failure probability

Simplified proof using hypergraph tournaments and information theory

Abstract

Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in NP/poly occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a streamlined proof of Drucker's theorem. An AND-compression is a deterministic polynomial-time algorithm that maps a set of SAT-instances $x_1,\dots,x_t$ to a single SAT-instance $y$ of size poly(max $|x_i|$) such that $y$ is satisfiable if and only if all $x_i$ are satisfiable. The "AND" in the name stems from the fact that the predicate "$y$ is satisfiable" can be written as the AND of all predicates "$x_i$ is satisfiable". Drucker's result complements the result by Bodlaender et al. (2009) and Fortnow and Santhanam (2010), who proved the analogous statement for OR-compressions, and Drucker's proof not only subsumes that result but also extends it to randomized compression algorithms that are allowed to have a certain probability of failure. Drucker (2012) presented two proofs: The first uses information theory and the minimax theorem from game theory, and the second is an elementary, iterative proof that is not as general. In our proof, we realize the iterative structure as a generalization of the arguments of Ko (1983) for P-selective sets, which use the fact that tournaments have dominating sets of logarithmic size. We generalize this fact to hypergraph tournaments. Our proof achieves the full generality of Drucker's theorem, avoids the minimax theorem, and restricts the use of information theory to a single, intuitive lemma about the average noise sensitivity of compressive maps. To prove this lemma, we use the same information-theoretic inequalities as Drucker.