arXiv subjects

Pavel V. Smirnov

Publications and source records attributed to Pavel V. Smirnov.

3 recordsLinked to original sources

arXiv2023

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasability of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.

cs.DS↗

arXiv2020

Parameterized Algorithms for Power-Efficiently Connecting Wireless Sensor Networks: Theory and Experiments

We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless communication network: Given an edge-weighted $n$-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. On the negative side, we show that $o(\log n)$-approximating the difference $d$ between the optimal solution cost and a natural lower bound is NP-hard and that, under the Exponential Time Hypothesis, there are no exact algorithms running in $2^{o(n)}$ time or in $f(d)\cdot n^{O(1)}$ time for any computable function $f$. Moreover, we show that the special case of connecting $c$ network components with minimum additional cost generally cannot be polynomial-time reduced to instances of size $c^{O(1)}$ unless the polynomial-time hierarchy collapses. On the positive side, we provide an algorithm that reconnects $O(\log n)$ connected components with minimum additional cost in polynomial time. These algorithms are motivated by application scenarios of monitoring areas or where an existing sensor network may fall apart into several connected components due to sensor faults. In experiments, the algorithm outperforms CPLEX with known ILP formulations when $n$ is sufficiently large compared to $c$.

cs.DS↗

arXiv2020

Optimal-size problem kernels for $d$-Hitting Set in linear time and space

The known linear-time kernelizations for $d$-Hitting Set guarantee linear worst-case running times using a quadratic-size data structure (that is not fully initialized). Getting rid of this data structure, we show that problem kernels of asymptotically optimal size $O(k^d)$ for $d$-Hitting Set are computable in linear time and space. Additionally, we experimentally compare the linear-time kernelizations for $d$-Hitting Set to each other and to a classical data reduction algorithm due to Weihe.

cs.DS↗