Parameterized Approximation via Fidelity Preserving Transformations

Abstract

Abstract.We motivate and describe a new parameterized approximation paradigm which studies the interaction between performance ratio and running time for any parameterization of a given optimization problem. As a key tool, we introduce the concept of α-shrinking transformation, for α ≥ 1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving approximation ratio of α (or α-fidelity). For example, it is well-known that Vertex Cover cannot be approximated within any constant factor better than 2 [22] (under usual assumptions). Our parameterized α-approximation algorithm for k-Vertex Cover, parameterized by the solution size, has a running time of 1.273(2−α)k , where the running time of the best FPT algorithm is 1.273 k [10]. Our algorithms define a continuous tradeoff between running times and approximation ratios, allowing practitioners to appropriately allocate computational resources. Moving even beyond the performance ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain kernels which are smaller than the best known for a given problem. For the Vertex Cover problem we obtain a kernel size of 2(2 − α)k. The smaller "α-fidelity" kernels allow us to solve exactly problem instances more efficiently, while obtaining an approximate solution for the original instance. We show that such transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree. We note that most of our algorithms are easy to implement and are therefore practical in use.

Lossy kernelization

“…The observation that a lossy pre-processing can simultaneously achieve a better size bound than normal kernelization algorithms as well as a better approximation factor than the ratio of the best approximation algorithms is not new. In particular, motivated by this observation Fellows et al [30] initiated the study of lossy kernelization. Fellows et al [30] proposed a definition of lossy kernelization called α-fidelity kernels.…”

“…In particular, motivated by this observation Fellows et al [30] initiated the study of lossy kernelization. Fellows et al [30] proposed a definition of lossy kernelization called α-fidelity kernels. Essentially, an α-fidelity kernel is a polynomial time pre-processing procedure such that an optimal solution to the reduced instance translates to an α-approximate solution to the original.…”

“…It is important to note that even though the definition of α-approximate kernels crucially Longest Path O( n log n ) [2] 2 (log n) 1− [39] no k O(1) [6] any α no k O (1) Set Cover/n ln n [55] (1 − ) ln n [47] no n O(1) [22] any α no n O (1) Hitting Set/n O( √ n) [48] 2 (log n) 1− [48] no n O(1) [22] any α no n O (1) Vertex Cover 2 [55] (2 − ) [21,41] 2k [15] 1 < α < 2 2(2 − α)k [30] d-Hitting Set d [55] d − [20,41]…”

Fractals for Kernelization Lower Bounds

“…Approximation algorithmics has a highly developed theory (having produced concepts such as MaxSNP-hardness and the famous PCP theory) for proving (relative to some plausible complexity-theoretic assumption) lower bounds on the approximation factors [1,44,46]. We remark that there exist frameworks combining kernelization and approximation algorithms, namely α-fidelity kernelization [22] and lossy kernelization [35].…”

New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set

“…This issue has already been considered for other FPT problems, in particular for the min vertex cover problem. In [2,3,13] several parameterized approximation algorithms running faster than (exact) FPT algorithms and achieving ratios better than the ratio 2 (achievable in polynomial time) are given. Note that [3,13] ask as open question if similar results can be achieved for edge dominating set.…”