Lower Bounds for Kernelization

Bodlaender, Hans L.

doi:10.1007/978-3-319-13524-3_1

Cited by 2 publications

(1 citation statement)

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“…For many problems, the search for small kernels is ongoing research, but not all problems admit such. This led to an approach that tries to disprove the existence of kernels of a certain size, see [15,17,16,30]. For the next theorem, we apply techniques developed in that line of work.…”

Section: Lower Bound On the Kernelmentioning

confidence: 99%

Fast Witness Counting

Chini,

Massoud,

Meyer

et al. 2018

Preprint

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We study the witness-counting problem: given a set of vectors V in the d-dimensional vector space over F 2 , a target vector t, and an integer k, count all ways to sum-up exactly k different vectors from V to reach t. The problem is well-known in coding theory and received considerable attention in complexity theory. Recently, it appeared in the context of hardware monitoring.Our contribution is an algorithm for witness counting that is optimal in the sense of finegrained complexity. It runs in time O ˚p2 d q with only a logarithmic dependence on m " |V |. The algorithm makes use of the Walsh-Hadamard transform to compute convolutions over F d 2 . The transform, however, overcounts the solutions. Inspired by the inclusion-exclusion principle, we introduce correction terms. The correction leads to a recurrence that we show how to solve efficiently. The correction terms are obtained from equivalence relations over F d 2 . We complement our upper bound with two lower bounds on the problem. The first relies on #ETH and prohibits an 2 opdq -time algorithm. The second bound states the non-existence of a polynomial kernel for the decision version of the problem.

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Section: Lower Bound On the Kernelmentioning

confidence: 99%