Approximation Schemes for Maximum Weight Independent Set of Rectangles

In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping rectangles. Due to many applications, e.g. in data mining, map labeling and admission control, the problem has received a lot of attention by various research communities. We present the first (1+epsilon)-approximation algorithm for the MWISR problem with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the best known polynomial time approximation algorithms for the problem achieve superconstant approximation ratios of O(log log n) (unweighted case) and O(log n / log log n) (weighted case). Key to our results is a new geometric dynamic program which recursively subdivides the plane into polygons of bounded complexity. We provide the technical tools that are needed to analyze its performance. In particular, we present a method of partitioning the plane into small and simple areas such that the rectangles of an optimal solution are intersected in a very controlled manner. Together with a novel application of the weighted planar graph separator theorem due to Arora et al. this allows us to upper bound our approximation ratio by (1+epsilon). Our dynamic program is very general and we believe that it will be useful for other settings. In particular, we show that, when parametrized properly, it provides a polynomial time (1+epsilon)-approximation for the special case of the MWISR problem when each rectangle is relatively large in at least one dimension. Key to this analysis is a method to tile the plane in order to approximately describe the topology of these rectangles in an optimal solution. This technique might be a useful insight to design better polynomial time approximation algorithms or even a PTAS for the MWISR problem

A QPTAS for Maximum Weight Independent Set of Polygons with Polylogarithmically Many Vertices

The Maximum Weight Independent Set of Polygons problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the 2-dimensional plane, the goal is to find a set of pairwise non-overlapping polygons with maximum total weight. Due to its wide range of applications, the MWISP problem and its special cases have been extensively studied both in the approximation algorithms and the computational geometry community. Despite a lot of research, its general case is not well-understood. Currently the best known polynomial time algorithm achieves an approximation ratio of n^(epsilon) [Fox and Pach, SODA 2011], and it is not even clear whether the problem is APX-hard. We present a (1+epsilon)-approximation algorithm, assuming that each polygon in the input has at most a polylogarithmic number of vertices. Our algorithm has quasi-polynomial running time. We use a recently introduced framework for approximating maximum weight independent set in geometric intersection graphs. The framework has been used to construct a QPTAS in the much simpler case of axis-parallel rectangles. We extend it in two ways, to adapt it to our much more general setting. First, we show that its technical core can be reduced to the case when all input polygons are triangles. Secondly, we replace its key technical ingredient which is a method to partition the plane using only few edges such that the objects stemming from the optimal solution are evenly distributed among the resulting faces and each object is intersected only a few times. Our new procedure for this task is not more complex than the original one, and it can handle the arising difficulties due to the arbitrary angles of the polygons. Note that already this obstacle makes the known analysis for the above framework fail. Also, in general it is not well understood how to handle this difficulty by efficient approximation algorithms

Towards a perfect fixed point action for SU(3) gauge theory

We present an overview of the construction and testing of actions for SU(3) gauge theory which are approximate fixed points of renormalization group equations (at $\beta\rightarrow \infty$). Such actions are candidates for use in numerical simulations on coarse lattices.Comment: 6 pages, uuencoded compressed postscript file, contribution to LAT9

Nature’s Lab for Derivatization: New and Revised Structures of a Variety of Streptophenazines Produced by a Sponge-Derived Streptomyces Strain

Eight streptophenazines (A–H) have been identified so far as products of Streptomyces strain HB202, which was isolated from the sponge Halichondria panicea from the Baltic Sea. The variation of bioactivities based on small structural changes initiated further studies on new derivatives. Three new streptophenazines (I–K) were identified after fermentation in the present study. In addition, revised molecular structures of streptophenazines C, D, F and H are proposed. Streptophenazines G and K exhibited moderate antibacterial activity against the facultative pathogenic bacterium Staphylococcus epidermidis and against Bacillus subtilis. All tested compounds (streptophenazines G, I–K) also showed moderate activities against PDE 4B

A component of the mitochondrial outer membrane proteome of T. brucei probably contains covalent bound fatty acids

A subclass of eukaryotic proteins is subject to modification with fatty acids, the most common of which are palmitic and myristic acid. Protein acylation allows association with cellular membranes in the absence of transmembrane domains. Here we examine POMP39, a protein previously described to be present in the outer mitochondrial membrane proteome (POMP) of the protozoan parasite Trypanosoma brucei. POMP39 lacks canonical transmembrane domains, but is likely both myristoylated and palmitoylated on its N-terminus. Interestingly, the protein is also dually localized on the surface of the mitochondrion as well as in the flagellum of both insect-stage and the bloodstream form of the parasites. Upon abolishing of global protein acylation or mutation of the myristoylation site, POMP39 relocates to the cytosol. RNAi-mediated ablation of the protein neither causes a growth phenotype in insect-stage nor bloodstream form trypanosomes

Phonological Factors Affecting L1 Phonetic Realization of Proficient Polish Users of English

Acoustic phonetic studies examine the L1 of Polish speakers with professional level proficiency in English. The studies include two tasks, a production task carried out entirely in Polish and a phonetic code-switching task in which speakers insert target Polish words or phrases into an English carrier. Additionally, two phonetic parameters are studied: the oft-investigated VOT, as well as glottalization vs. sandhi linking of word-initial vowels. In monolingual Polish mode, L2 interference was observed for the VOT parameter, but not for sandhi linking. It is suggested that this discrepancy may be related to the differing phonological status of the two phonetic parameters. In the code-switching tasks, VOTs were on the whole more English-like than in monolingual mode, but this appeared to be a matter of individual performance. An increase in the rate of sandhi linking in the code-switches, except for the case of one speaker, appeared to be a function of accelerated production of L1 target items

Fixed point actions for SU(3) gauge theory

We summarize our recent work on the construction and properties of fixed point (FP) actions for lattice $SU(3)$ pure gauge theory. These actions have scale invariant instanton solutions and their spectrum is exact through 1--loop, i.e. in their physical predictions there are no $a^n$ nor $g^2 a^n$ cut--off effects for any $n$. We present a few-parameter approximation to a classical FP action which is valid for short correlation lengths. We perform a scaling test of the action by computing the quantity $G = L \sqrt{\sigma(L)}$, where the string tension $\sigma(L)$ is measured from the torelon mass $\mu = L \sigma(L)$, on lattices of fixed physical volume and varying lattice spacing $a$. While the Wilson action shows scaling violations of about ten per cent, the approximate fixed point action scales within the statistical errors for $1/2 \ge aT_c$.Comment: 11 pages, uuencoded compressed postscript fil

The classically perfect fixed point action for SU(3) gauge theory

In this paper (the first of a series) we describe the construction of fixed point actions for lattice $SU(3)$ pure gauge theory. Fixed point actions have scale invariant instanton solutions and the spectrum of their quadratic part is exact (they are classical perfect actions). We argue that the fixed point action is even 1--loop quantum perfect, i.e. in its physical predictions there are no $g^2 a^n$ cut--off effects for any $n$. We discuss the construction of fixed point operators and present examples. The lowest order $q {\bar q}$ potential $V(\vec{r})$ obtained from the fixed point Polyakov loop correlator is free of any cut--off effects which go to zero as an inverse power of the distance $r$.Comment: 34 pages (latex) + 7 figures (Postscript), uuencode