Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at http://arxiv.org/abs/1410.220

Natural hybridization between Populus nigra L. and P. x canadensis Moench. Hybrid offspring competes for niches along the Rhine river in the Netherlands

Black poplar (Populus nigra L.) is a major species for European riparian forests but its abundance has decreased over the decades due to human influences. For restoration of floodplain woodlands, the remaining black poplar stands may act as source population. A potential problem is that P. nigra and Populus deltoides have contributed to many interspecific hybrids, which have been planted in large numbers. As these Populus x canadensis clones have the possibility to intercross with wild P. nigra trees, their offspring could establish themselves along European rivers. In this study, we have sampled 44 poplar seedlings and young trees that occurred spontaneously along the Rhine river and its tributaries in the Netherlands. Along these rivers, only a few native P. nigra L. populations exist in combination with many planted cultivated P. x canadensis trees. By comparison to reference material from P. nigra, P. deltoides and P. x canadensis, species-specific AFLP bands and microsatellite alleles indicated that nearly half of the sampled trees were not pure P. nigra but progeny of natural hybridisation that had colonised the Rhine river banks. The posterior probability method as implemented in NewHybrids using microsatellite data was the superior method in establishing the most likely parentage. The results of this study indicate that offspring of hybrid cultivated poplars compete for the same ecological niche as native black poplars

The Traveling Salesman Problem

This paper presents a self-contained introduction into algorithmic and computational aspects of the traveling salesman problem and of related problems, along with their theoretical prerequisites as seen from the point of view of an operations researcher who wants to solve practical problem instances. Extensive computational results are reported on most of the algorithms described. Optimal solutions are reported for instances with sizes up to several thousand nodes as well as heuristic solutions with provably very high quality for larger instances

Polyhedral techniques in combinatorial optimization II: computations

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Genetic Diversity, Phylogenetics and Molecular Systematics of Guizotia Cass. (Asteraceae)

The genus Guizotia belongs to the tribe Heliantheae in the family Asteraceae. It has been placed under different subtribes. The genus has its center of origin, distribution and genetic diversity in Ethiopia, where G. abyssinica (niger) has been domesticated. Amplified Fragment Length Polymorphism (AFLP), Random Amplified Polymorphic DNA (RAPD) and DNA sequencing were applied to study the genetic diversity, phylogenetics, and molecular systematics of this genus. A large number of niger populations, representing all regions in Ethiopia where this crop is grown, was investigated using AFLP and RAPD molecular marker techniques. The extent of genetic variation in niger is distributed throughout its growing regions, regardless of the extent and altitude of cultivation. Despite the fact that most of the variation was within populations, significant population differentiation was obtained (AMOVA; P < 0.001) in all guizotias. It is concluded that both G. abyssinica and its wild and/or weedy relatives have wide genetic bases that need to be conserved and utilized for the improvement of G. abyssinica. Further collection of niger germplasm and exploration and conservation of highly localized guizotias are recommended. Most of the diagnostic markers generated from AFLPs and RAPDs in this study were specific to G. arborescens and G. zavattarii. Phylogenetic analyses of the genus Guizotia were undertaken based on molecular sequence data from the internal transcribed spacers (ITS) and five chloroplast DNA regions. The analyses revealed a close phylogenetic relationship between G. abyssinica and G. scabra ssp. schimperi and support the previous suggestion that the latter is the progenitor of the former. According to this study, G. scabra ssp. scabra, G. scabra ssp. schimperi, and the Chelelu and Ketcha populations are best viewed at present as separate species within the genus Guizotia. Those perennial guizotias with highly localized geographic distribution appears to have evolved first during the evolutionary history of the genus. This study supports the placement of the genus Guizotia within the subtribe Milleriinae. It is suggested that the present species composition of Guizotia and the subtribal placement of the genus need to be redefined

Spartan Daily, January 31, 1997

Volume 108, Issue 6https://scholarworks.sjsu.edu/spartandaily/9083/thumbnail.jp