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Revision #3 to TR14-075 | 23rd September 2014 17:10

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

Revision #3
Authors: Holger Dell
Accepted on: 23rd September 2014 17:10
Keywords:

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.

Changes to previous version:

fixed weird citation style.

Revision #2 to TR14-075 | 23rd September 2014 16:20

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

Revision #2
Authors: Holger Dell
Accepted on: 23rd September 2014 16:20
Keywords:

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.

Changes to previous version:

Changed title, abstract, and introduction.

Revision #1 to TR14-075 | 17th May 2014 19:04

#### A simple proof that AND-compression of NP-complete problems is hard

Revision #1
Authors: Holger Dell
Accepted on: 17th May 2014 19:04
Keywords:

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 simple proof of this result.

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.

The overall structure of our proof is similar to 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. For the information-theoretic part of the proof, we consider a natural generalization of the average noise sensitivity of a Boolean function, which is bounded for compressive maps. We prove this with mechanical calculations that involve the Kullback-Leibler divergence.

Changes to previous version:

fixed weird citation style.

### Paper:

TR14-075 | 16th May 2014 18:05

#### A simple proof that AND-compression of NP-complete problems is hard

TR14-075
Authors: Holger Dell
Publication: 17th May 2014 18:39
Keywords:

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 simple proof of this result.

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.

The overall structure of our proof is similar to 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. For the information-theoretic part of the proof, we consider a natural generalization of the average noise sensitivity of a Boolean function, which is bounded for compressive maps. We prove this with mechanical calculations that involve the Kullback-Leibler divergence.

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