The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+eps)-approximations in f(k,eps)n^ O(1) time where k is some parameter of the input. We obtain the following results on parameterized approximability: 1) In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(loglog n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant eps>0 and integer k>0, in time f(k,eps)n^ g(eps) , either outputs a solution of size at least k (1+eps), or declares that the optimum solution has size less than k. 2) In the (2-dimensional) geometric knapsack problem (TDK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of TDK with rotations (TDKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factors are 558 325+eps and 4 3+eps, resp. [, FOCS'17]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for TDKR. For all considered problems, getting time f(k,eps)n^ O(1) , rather than f(k,eps)n^ g(eps) , would give FPT time f'(k)n^ O(1) exact algorithms using eps=1 (k+1), contradicting W[1]-hardness.
For the special case of where all input objects are squares a PTAS is known @cite_16 but there can be no EPTAS @cite_27 . Recently, @cite_1 found polynomial-time algorithms for and with approximation ratio smaller than @math (also for the weighted case). For the special case that all input objects are squares there is a PTAS @cite_19 and even an EPTAS @cite_15 .
@cite_1 @cite_19 @cite_27 @cite_15 @cite_16
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An EPTAS (efficient PTAS) is an approximation scheme where e does not appear in the exponent of n, i.e., the running time is f(e)nc. We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answering an open question of Alber and Fiala [2], we show that Maximum Independent Set is W[1]-complete for the intersection graphs of unit disks and axis-parallel unit squares in the plane. A standard consequence of this result is that the @math time PTAS of [11] for Maximum Independent Set on unit disk graphs cannot be improved to an EPTAS. Similar results are obtained for the problem of covering points with squares. Purpose – The purpose of this paper is to suggest an alternative to the widely used Times Cited criterion for analysing citation networks. The approach involves taking account of the natures of the papers that cite a given paper, so as to differentiate between papers that attract the same number of citations.Design methodology approach – ArticleRank is an algorithm that has been derived from Google's PageRank algorithm to measure the influence of journal articles. ArticleRank is applied to two datasets – a citation network based on an early paper on webometrics, and a self‐citation network based on the 19 most cited papers in the Journal of Documentation – using citation data taken from the Web of Knowledge database.Findings – ArticleRank values provide a different ranking of a set of papers from that provided by the corresponding Times Cited values, and overcomes the inability of the latter to differentiate between papers with the same numbers of citations. The difference in rankings between Times Cited ... Given n length-L strings S = s1, …, s n over a constant size alphabet Σ together with an integer l, where l ≤ L, the objective of Consensus Patterns is to find a length-l string s, a substring t i of each s i in S such that ∑ ∀ id(t i , s) is minimized. Here d(x, y) denotes the Hamming distance between the two strings x and y. Consensus Patterns admits a PTAS [, JCSS 2002] is fixed parameter tractable when parameterized by the objective function value [Marx, SICOMP 2008], and although it is a well-studied problem, improvement of the PTAS to an EPTAS seemed elusive. We prove that Consensus Patterns does not admit an EPTAS unless FPT=W[1], answering an open problem from [, STACS 2002, Combinatorica 2006]. To the best of our knowledge, Consensus Patterns is the first problem that admits a PTAS, and is fixed parameter tractable when parameterized by the value of the objective function but does not admit an EPTAS under plausible complexity assumptions. The proof of our hardness of approximation result combines parameterized reductions and gap preserving reductions in a novel manner. Abstract Documents in computer-readable form can be used to provide information about other documents, i.e. those they cite. To do this efficiently requires procedures for computer recognition of citing statements. This is not easy, especially for multi-sentence citing statements. Computer recognition procedures have been developed which are accurate to the following extent: 73 of the words in statements selected by computer procedures as being citing statements are words which are correctly attributable to the corresponding documents. The retrieval effectiveness of computer-recognized citing statements was tested in the following way. First, for eight retrieval requests in inorganiic chemistry, average recall by search of Chemical Abstracts Service indexing and Chemical Abstracts abstract text words was found to be 50 . Words from citing statements referring to the papers to be retrieved were then added to the index terms and abstract words as additional access points, and searching was repeated. Average recall increased to 70 . Only words from citing statements published within a year of the cited papers were used. The retrieval effect of citing statement words alone (published within a year) without index or abstract terms was the following: average recall was 40 . When just the words of the titles of the cited papers were added to those citing statement words, average recall increased to 50 .