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Orientation of matrices
Abstract
Camion proved that every real-valued matrix A can be transformed by pivoting operations and nonzero multiplications of columns into a nonnegative matrix. In this paper
we describe a finite algorithm to make this transformation, based on the results of Camion. Our main result is that when A is a totally unimodular matrix this transformation can be made by a polynomial algorithm.
Key wordsLinear Algebra–Matroid Theory–Total Unimodularity
... There is no known polynomial-time algorithm to find a Camion basis in general. Fonlupt and Raco [24] described a finite procedure to find one based on the results of Camion. They also gave an algorithm which runs in time O(n 3 m 2 ) for totally unimodular matrices. ...
... We will see an algorithm, called Simp, that runs in O(∆ 3 (n m) 2 ) if M is an integral matrix, where ∆ is the greatest determinant (in absolute value) of a basis. So, for the particular case of totally unimodular matrices (where ∆ = 1), Simp is faster than the algorithm of Fonlupt and Raco [24]. Moreover, the procedure Simp applied to real matrices is also finite. ...
In this thesis, we deal with binet matrices, an extension of network matrices. The main result of this thesis is the following. A rational matrix A of size n×m can be tested for being binet in time O(n6m). If A is binet, our algorithm outputs a nonsingular matrix B and a matrix N such that [B N] is the node-edge incidence matrix of a bidirected graph (of full row rank) and A = B-1N. Furthermore, we provide some results about Camion bases. For a matrix M of size n × m', we present a new characterization of Camion bases of M, whenever M is the node-edge incidence matrix of a connected digraph (with one row removed). Then, a general characterization of Camion bases as well as a recognition procedure which runs in O(n2m') are given. An algorithm which finds a Camion basis is also presented. For totally unimodular matrices, it is proven to run in time O((nm)2) where m = m' – n. The last result concerns specific network matrices. We give a characterization of nonnegative {ε, ρ}-noncorelated network matrices, where ε and ρ are two given row indexes. It also results a polynomial recognition algorithm for these matrices.
... There is no known polynomial-time algorithm to find a Camion basis in general. Fonlupt and Raco [2] described a finite procedure to find one based on the results of Camion. They also gave an algorithm which runs in time O(n 3 m 2 ) for totally unimodular matrices. ...
Let M be a finite set of vectors in Rn of cardinality m and H(M)={{x∈Rn:cTx=0}:c∈M} the central hyperplane arrangement represented by M. An independent subset of M of cardinality n is called a Camion basis, if it determines a simplex region in the arrangement H(M). In this paper, we first present a new characterization of Camion bases, in the case where M is the column set of the node-edge incidence matrix (without one row) of a given connected digraph. Then, a general characterization of Camion bases as well as a recognition procedure which runs in O(n2m) are given. Finally, an algorithm which finds a Camion basis is presented. For certain classes of matrices, including totally unimodular matrices, it is proven to run in polynomial time and faster than the algorithm due to Fonlupt and Raco.
Binet matrices generalize network matrices and play an important role in combinatorial optimization. A first polynomial-time
algorithm for recognizing binet matrices appeared in the author’s doctoral thesis. In this paper, we present some key ideas
and results involved in the design of this algorithm. We show how we can find a Camion basis of the input matrix, whenever
this one is binet, and then reduce the recognition problem to that of special binet matrices called bicyclic and cyclic.
Damaschke, Müller, and Kratsch [Inform. Process. Lett., 36 (1990), pp. 231--236] gave a polynomial-time algorithm to solve the minimum dominating set problem in convex bipartite graphs , that is, where the nodes in Y can be ordered so that each node of X is adjacent to a contiguous sequence of nodes. Gamble et al. [Graphs Combin., 11 (1995), pp. 121--129] gave an extension of their algorithm to weighted dominating sets. We formulate the dominating set problem as that of finding a minimum weight subset of elements of a graphic matroid, which covers each fundamental circuit and fundamental cut with respect to some spanning tree T. When T is a directed path, this simultaneous covering problem coincides with the dominating set problem for the previously studied class of convex bipartite graphs. We describe a polynomial-time algorithm for the more general problem of simultaneous covering in the case when T is an arborescence. We also give NP-completeness results for fairly specialized classes of the simultaneous cover problem. These are based on connections between the domination and induced matching problems.
L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method
is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method
also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in
perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for
the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields
a proof that weighted fractional chromatic number is NP-hard.
A cocircuit of a matroid is separating if deleting it leaves a separable matroid. We give an effecient algorithm which finds a separating cocircuit or a Fano minor in a binary matroid, thus proving constructively a theorem of Tutte. Using this algorithm and a new recursive characterization of bond matroids, we give a new method for testing binary matroids for graphicness. We also give an efficient algorithm for finding a special kind of separating cocircuit: one whose deletion leaves a matroid having a coloop.
It is proved that every regular matroid may be constructed by piecing together graphic and cographic matroids and copies of a certain 10-element matroid.
Lecutres on matroids
- W T Tutte
- W.T. Tutte