Approximating Semi-Matchings in Streaming and in Two-Party Communication

We study the communication complexity and streaming complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph G = (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 <= \epsilon <= 1 uses space O(n^{1+\epsilon}) and computes an O(n^{(1-\epsilon)/2})-approximation to the semi-matching problem. Furthermore, with O(log n) passes it is possible to compute an O(log n)-approximation with space O(n). In the one-way two-party communication setting, we show that for every \epsilon > 0, deterministic communication protocols for computing an O(n^{1/((1+\epsilon)c + 1)})-approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2n edges that compute an O(sqrt(n)) and an O(n^{1/3})-approximation respectively. Finally, we improve on results of Harvey et al. [Journal of Algorithms 2006] and prove new links between semi-matchings and matchings. While it was known that an optimal semi-matching contains a maximum matching, we show that there is a hierarchical decomposition of an optimal semi-matching into maximum matchings. A similar result holds for semi-matchings that do not admit length-two degree-minimizing paths.Comment: This is the long version including all proves of the ICALP 2013 pape

On the Power of Advice and Randomization for Online Bipartite Matching

While randomized online algorithms have access to a sequence of uniform random bits, deterministic online algorithms with advice have access to a sequence of advice bits, i.e., bits that are set by an all powerful oracle prior to the processing of the request sequence. Advice bits are at least as helpful as random bits, but how helpful are they? In this work, we investigate the power of advice bits and random bits for online maximum bipartite matching (MBM). The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized $(1-\frac{1}{e})$-competitive algorithm for \textsc{MBM} that requires access to $\Theta(n \log n)$ uniform random bits. We show that $\Omega(\log(\frac{1}{\epsilon}) n)$ advice bits are necessary and $O(\frac{1}{\epsilon^5} n)$ sufficient in order to obtain a $(1-\epsilon)$-competitive deterministic advice algorithm. Furthermore, for a large natural class of deterministic advice algorithms, we prove that $\Omega(\log \log \log n)$ advice bits are required in order to improve on the $\frac{1}{2}$-competitiveness of the best deterministic online algorithm, while it is known that $O(\log n)$ bits are sufficient. Last, we give a randomized online algorithm that uses $c n$ random bits, for integers $c \ge 1$, and a competitive ratio that approaches $1-\frac{1}{e}$ very quickly as $c$ is increasing. For example if $c = 10$, then the difference between $1-\frac{1}{e}$ and the achieved competitive ratio is less than $0.0002$

IFCN Cash Crop: Benchmarking Farms Globally Oilseed Production Costs

Vegetable oil production has become one of the fastest expanding cash crop sectors in the last 50 years and it is still increasing rapidly. However the regions of expansion, the sources of plant oil and their importance vary over time. To shed light on this development it is necessary to look at the farm level production systems and their production costs for a variety of countries and oilseeds. In this paper we present the first results of the IFCN Cash Crop Network covering the international comparison of oilseed producing farms. A total of 25 farms with oilseed production have been analysed in compiling this paper. All farms produce at least one of the following oilseeds: soybeans, oilseed rape, sunflower and (two farms) mustard. The farms are located in 14 different countries/regions and represent typical oilseed producing farms in their region/country. The farm data was collected and compiled in all countries and regions by IFCN Partners according to IFCN standards to en-sure its international comparability. The most competitive farms in oilseed production worldwide can be found in South America. The farms in Ukraine also have great potential. At the moment the farms in Argentina show the highest profit margins. The farms from North America can also cover their full costs with the prices they receive for soybeans, sunflower and rape seed.Farm comparisons, Cash Crop production, International competitiveness, International Farm Comparison Network, Oilseed, Benchmarking, Crop Production/Industries,

Tricritical behavior of the massive chiral Gross-Neveu model

The phase diagram of the massive chiral Gross-Neveu model (the 1+1-dimensional Nambu-Jona-Lasinio model at large N) is investigated in the vicinity of the tricritical point. Using the derivative expansion, the grand canonical potential is cast into the form of a Ginzburg-Landau effective action. Minimization of this action by variational and numerical methods reveals both 1st and 2nd order phase transitions to a chiral crystal phase, separated by a tricritical line. These findings are contrasted to the massive Gross-Neveu model with discrete chiral symmetry where only 2nd order transitions have been observed.Comment: 10 pages, 10 figures; v2: More details about perturbation theory given, cf Eqs. (46-48