We consider polynomials f (x(1),..., x(n)) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f (x1,..., x(n)) = b coincides with the set of s...
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We consider polynomials f (x(1),..., x(n)) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f (x1,..., x(n)) = b coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with respect to the right-hand side b. We obtain the properties of multiaffine polynomials over a finite field. Then we show that checking the multiaffinity with respect to a given right-hand side may be done by an algorithm with polynomial (in terms of the number of variables and summands of the input polynomial) complexity. Besides that, we prove that in case of the positive decision a corresponding system of linear equations may be recovered with complexity which is also polynomial in terms of the same parameters.
Batching problems are combinations of sequencing and partitioning problems. For each job sequence JS there is a partition of JS into batches with optimal value opt(JS) and one has to find a job sequence which minimize...
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Batching problems are combinations of sequencing and partitioning problems. For each job sequence JS there is a partition of JS into batches with optimal value opt(JS) and one has to find a job sequence which minimizes this optimal value. We show that in many situations opt(JS) is the solution of a shortest path problem in some network. An algorithm solving this special shortest path problem in linear time with respect to the number of vertices is presented. Using this algorithm some new batching results are derived. Furthermore. it is shown that most of the batching problems which are known to be polynomially solvable turn into NP-hard problems when modified slightly.
In the article, we consider undirected multiple graphs of any natural multiplicity k > 1. A multiple graph contains edges of three types: ordinary edges, multiple edges, and multiedges. Each edge of the last two ty...
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In the article, we consider undirected multiple graphs of any natural multiplicity k > 1. A multiple graph contains edges of three types: ordinary edges, multiple edges, and multiedges. Each edge of the last two types is a union of k linked edges, which connect 2 or (k + 1) vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, then it can be incident to other multiple edges, and it can also be the common end of k linked edges of a multiedge. If a vertex is the common end of a multiedge, then it cannot be the common end of another multiedge. As for an ordinary graph, we can define the integer function of the length of an edge for a multiple graph and set the problem of the shortest path joining two vertices. Any multiple path is a union of k ordinary paths adjusted on the linked edges of all multiple and multiedge edges. In this article, the previously obtained algorithm for finding the shortest path in an arbitrary multiple graph is optimized. We show that the optimized algorithm is polynomial. Thus, the shortest path problem is polynomial for any multiple graph.
A graph is said to be (p,q)-colorable if its vertex set can be partitioned into at most p cliques and q independent sets. In particular, (0, 2)-colorable graphs are bipartite, and (1, 1)-colorable are the split graphs...
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A graph is said to be (p,q)-colorable if its vertex set can be partitioned into at most p cliques and q independent sets. In particular, (0, 2)-colorable graphs are bipartite, and (1, 1)-colorable are the split graphs. For both of these classes, the problem of finding a maximum weight independent set is known to be solvable in polynomial time. In the present note, we give a complete classification of the family of (p, q)-colorable graphs with respect to time complexity of this problem. Specifically, we show that the problem has a polynomial time solution in the class of (p, q)-colorable graphs if and only if q less than or equal to 2 (assuming P not equal NP). (C) 2003 Elsevier Science B.V. All rights reserved.
Polar, monopolar, and unipolar graphs are defined in terms of the existence of certain vertex partitions. Although it is polynomial to determine whether a graph is unipolar and to find whenever possible a unipolar par...
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Polar, monopolar, and unipolar graphs are defined in terms of the existence of certain vertex partitions. Although it is polynomial to determine whether a graph is unipolar and to find whenever possible a unipolar partition, the problems of recognizing polar and monopolar graphs are both NP-complete in general. These problems have recently been studied for chordal, claw-free, and permutation graphs. polynomial time algorithms have been found for solving the problems for these classes of graphs, with one exception: polarity recognition remains NP-complete in claw-free graphs. In this paper, we connect these problems to edge-coloured homomorphism problems. We show that finding unipolar partitions in general and finding monopolar partitions for certain classes of graphs can be efficiently reduced to a polynomial-time solvable 2-edge-coloured homomorphism problem, which we call the colour-bipartition problem. This approach unifies the currently known results on monopolarity and extends them to new classes of graphs.
A Halin graphH=T∪C is obtained by embedding a treeT having no nodes of degree 2 in the plane, and then adding a cycleC to join the leaves ofT in such a way that the resulting graph is planar. These graphs are edge mi...
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A Halin graphH=T∪C is obtained by embedding a treeT having no nodes of degree 2 in the plane, and then adding a cycleC to join the leaves ofT in such a way that the resulting graph is planar. These graphs are edge minimal 3-connected, hamiltonian, and in general have large numbers of hamilton cycles. We show that for arbitrary real edge costs the travelling salesman problem can be polynomially solved for such a graph, and we give an explicit linear description of the travelling salesman polytope (the convex hull of the incidence vectors of the hamilton cycles) for such a graph.
Contaminant oligonucleotide sequences such as primers and adapters can occur in both ends of high-throughput sequencing (HTS) reads. ALIENTRIMMER was developed in order to detect and remove such contaminants. Based on...
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Contaminant oligonucleotide sequences such as primers and adapters can occur in both ends of high-throughput sequencing (HTS) reads. ALIENTRIMMER was developed in order to detect and remove such contaminants. Based on the decomposition of specified alien nucleotide sequences into k-mers, ALIENTRIMMER is able to determine whether such alien k-mers are occurring in one or in both read ends by using a simple polynomial algorithm. Therefore, ALIENTRIMMER can process typical HTS single- or paired-end files with millions of reads in several minutes with very low computer resources. Based on the analysis of both simulated and real-case Illumina(R), 454(TM) and Ion Torrent(TM) read data, we show that ALIENTRIMMER performs with excellent accuracy and speed in comparison with other trimming tools. The program is freely available at ftp://***/pub/gensoft/projects/AlienTrimmer/. (C) 2013 Elsevier Inc. All rights reserved.
In this paper, we study cyclic production for throughput optimization in robotic flow-shops. We are focusing on simple production cycles. Robotic cells can have a linear or a circular layout: most classical results on...
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The reflector antenna design problem requires to solve a second boundary value problem for a complicated Monge-Ampere equation, for which the traditional discretization methods fail. In this paper we reduce the proble...
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The reflector antenna design problem requires to solve a second boundary value problem for a complicated Monge-Ampere equation, for which the traditional discretization methods fail. In this paper we reduce the problem to that of finding a minimizer or a maximizer of a linear functional subject to a linear constraint. Therefore it becomes an linear optimization problem and algorithms in linear programming apply.
Let G = (V, E) be a multigraph which has a designated vertex s is an element of V with an even degree. For two edges e(1) = (s, u(1)) and e(2) = (s, u(2)), we say that a multigraph G' is obtained from G by splitti...
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Let G = (V, E) be a multigraph which has a designated vertex s is an element of V with an even degree. For two edges e(1) = (s, u(1)) and e(2) = (s, u(2)), we say that a multigraph G' is obtained from G by splitting e(1) and e(2) at s if two edges e(1) and e(2) are replaced with a single edge (u(1), u(2)). It is known that all edges incident to s can be split without losing the edge-connectivity of G in V - s. This complete splitting plays an important role in solving many graph connectivity problems. The currently fastest algorithm for a complete splitting [14] runs in O(n(m + n log n) log n) time, where n = \V\ and m is the number of pairs of vertices between which G has an edge. Their algorithm is first designed for Eulerian multigraphs, and then extended for general multigraphs. Although the part for Eulerian multigraphs is simple, the rest for general multigraphs is considerably complicated. This paper proposes a much simpler O (n (m + n log n) log n) time algorithm for finding a complete splitting. A new edge-splitting theorem derived from our algorithm is also presented.
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