Approximation Bounds For Minimum Degree Matching

We consider the MINGREEDY strategy for Maximum Cardinality Matching. MINGREEDY repeatedly selects an edge incident with a node of minimum degree. For graphs of degree at most $\Delta$ we show that MINGREEDY achieves approximation ratio at least $\frac{\Delta-1}{2\Delta-3}$ in the worst case and that this performance is optimal among adaptive priority algorithms in the vertex model, which include many prominent greedy matching heuristics. Even when considering expected approximation ratios of randomized greedy strategies, no better worst case bounds are known for graphs of small degrees.Comment: % CHANGELOG % rev 1 2014-12-02 % - Show that the class APV contains many prominent greedy matching algorithms. % - Adapt inapproximability bound for APV-algorithms to a priori knowledge on |V|. % rev 2 2015-10-31 % - improve performance guarantee of MINGREEDY to be tigh

SuperSAGE: the drought stress-responsive transcriptome of chickpea roots

Background Drought is the major constraint to increase yield in chickpea (Cicer arietinum). Improving drought tolerance is therefore of outmost importance for breeding. However, the complexity of the trait allowed only marginal progress. A solution to the current stagnation is expected from innovative molecular tools such as transcriptome analyses providing insight into stress-related gene activity, which combined with molecular markers and expression (e)QTL mapping, may accelerate knowledge-based breeding. SuperSAGE, an improved version of the serial analysis of gene expression (SAGE) technique, generating genome-wide, high-quality transcription profiles from any eukaryote, has been employed in the present study. The method produces 26 bp long fragments (26 bp tags) from defined positions in cDNAs, providing sufficient sequence information to unambiguously characterize the mRNAs. Further, SuperSAGE tags may be immediately used to produce microarrays and probes for real-time-PCR, thereby overcoming the lack of genomic tools in non-model organisms. Results We applied SuperSAGE to the analysis of gene expression in chickpea roots in response to drought. To this end, we sequenced 80,238 26 bp tags representing 17,493 unique transcripts (UniTags) from drought-stressed and non-stressed control roots. A total of 7,532 (43%) UniTags were more than 2.7-fold differentially expressed, and 880 (5.0%) were regulated more than 8-fold upon stress. Their large size enabled the unambiguous annotation of 3,858 (22%) UniTags to genes or proteins in public data bases and thus to stress-response processes. We designed a microarray carrying 3,000 of these 26 bp tags. The chip data confirmed 79% of the tag-based results, whereas RT-PCR confirmed the SuperSAGE data in all cases. Conclusion This study represents the most comprehensive analysis of the drought-response transcriptome of chickpea available to date. It demonstrates that – inter alias – signal transduction, transcription regulation, osmolyte accumulation, and ROS scavenging undergo strong transcriptional remodelling in chickpea roots already 6 h after drought stress. Certain transcript isoforms characterizing these processes are potential targets for breeding for drought tolerance. We demonstrate that these can be easily accessed by micro-arrays and RT-PCR assays readily produced downstream of SuperSAGE. Our study proves that SuperSAGE owns potential for molecular breeding also in non-model crops

Degree heuristics for matching

Algorithms for the Maximum Cardinality Matching Problem which greedily add edges to the solution enjoy great popularity. We systematically study strengths and limitations of such algorithms, in particular of those which consider node degree information to select the next edge. Concentrating on nodes of small degree is a promising approach: it was shown, experimentally and analytically, that very good approximate solutions are obtained for restricted classes of random graphs. Results achieved under these idealized conditions, however, remained unsupported by statements which depend on less optimistic assumptions. The KarpSipser algorithm and 1-2-Greedy, which is a simplified variant of the well-known MinGreedy algorithm, proceed as follows. In each step, if a node of degree one (resp. at most two) exists, then an edge incident with a minimum degree node is picked, otherwise an arbitrary edge is added to the solution. We analyze the approximation ratio of both algorithms on graphs of degree at most D. Families of graphs are known for which the expected approximation ratio converges to 1/2 as D grows to infinity, even if randomization against the worst case is used. If randomization is not allowed, then we show the following convergence to 1/2: the 1-2-Greedy algorithm achieves approximation ratio (D-1)/(2D-3); if the graph is bipartite, then the more restricted KarpSipser algorithm achieves the even stronger factor D/(2D-2). These guarantees set both algorithms apart from other famous matching heuristics like e.g. Greedy or MRG: these algorithms depend on randomization to break the 1/2-barrier even for paths with D=2. Moreover, for any D our guarantees are strictly larger than the best known bounds on the expected performance of the randomized variants of Greedy and MRG. To investigate whether KarpSipser or 1-2-Greedy can be refined to achieve better performance, or be simplified without loss of approximation quality, we systematically study entire classes of deterministic greedy-like algorithms for matching. Therefore we employ the adaptive priority algorithm framework by Borodin, Nielsen, and Rackoff: in each round, an adaptive priority algorithm requests one or more edges by formulating their properties---like e.g. "is incident with a node of minimum degree"---and adds the received edges to the solution. No constraints on time and space usage are imposed, hence an adaptive priority algorithm is restricted only by its nature of picking edges in a greedy-like fashion. If an adaptive priority algorithm requests edges by processing degree information, then we show that it does not surpass the performance of KarpSipser: our D/(2D-2)-guarantee for bipartite graphs is tight and KarpSipser is optimal among all such "degree-sensitive" algorithms even though it uses degree information merely to detect degree-1 nodes. Moreover, we show that if degrees of both nodes of an edge may be processed, like e.g. the Double-MinGreedy algorithm does, then the performance of KarpSipser can only be increased marginally, if at all. Of special interest is the capability of requesting edges not only by specifying the degree of a node but additionally its set of neighbors. This enables an adaptive priority algorithm to "traverse" the input graph. We show that on general degree-bounded graphs no such algorithm can beat factor (D-1)/(2D-3). Hence our bound for 1-2-Greedy is tight and this algorithm performs optimally even though it ignores neighbor information. Furthermore, we show that an adaptive priority algorithm deteriorates to approximation ratio exactly 1/2 if it does not request small degree nodes. This tremendous decline of approximation quality happens for graphs on which 1-2-Greedy and KarpSipser perform optimally, namely paths with D=2. Consequently, requesting small degree nodes is vital to beat factor 1/2. Summarizing, our results show that 1-2-Greedy and KarpSipser stand out from known and hypothetical algorithms as an intriguing combination of both approximation quality and conceptual simplicity