Some lower bounds in parameterized ${\rm AC}^0$

Some Lower Bounds in Parameterized AC^0

We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical AC0. Among others, we derive such a lower bound for all fptapproximations of the parameterized clique problem and for a parameterized halting problem, which recently turned out to link problems of computational complexity, descriptive complexity, and proof theory. To show the fir...

Weak compositions and their applications to polynomial lower bounds for kernelization

We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1}∗ × N be two parameterized problems where the unparameterized version of L1 is NP-hard. Assuming coNP 6⊆ NP/poly, our framework essentially states that composing t L1-instances ...

A reduction of proof complexity to computational complexity for $AC^0[p]$ Frege systems

We give a general reduction of lengths-of-proofs lower bounds for constant depth Frege systems in DeMorgan language augmented by a connective counting modulo a prime p (the so called AC[p] Frege systems) to computational complexity lower bounds for search tasks involving search trees branching upon values of maps on the vector space of low degree polynomials over Fp. In 1988 Ajtai [2] proved th...

Spectral Methods for Matrix Rigidity with Applications

The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have implications to complexity theory. We consider restricted variants of the rigidity problem over the complex numbers. Using spectral methods, we ...

Some Lower Bounds for Estrada Index