Decomposable polymatroids and connections with graph coloring

We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of rank functions of matroids, and for which the minimum number of matroids required in such sums is the chromatic number of the line graph of the hypergraph. This result motivates introducing chromatic numbers and chromatic polynomials for polymatroids. We show that the chromatic polynomial of any 2-polymatroid is a rational multiple of the chromatic polynomial of some graph. We also find the excluded minors for the minor-closed class of polymatroids that can be written as sums of rank functions of matroids that form a chain of quotients.     

Efficient algorithms for a family of matroid intersection problems

Consider a matroid where each element has a real-valued cost and a color, red or green; a base is sought that contains q red elements and has smallest possible cost. An algorithm for the problem on general matroids is presented, along with a number of variations. Its efficiency is demonstrated by implementations on specific matroids. In all cases but one, the running time matches the best-known algorithm for the problem without the red element constraint: On graphic matroids, a smallest spanning tree with q red edges can be found in time O(n log n) more than what is needed to find a minimum spanning tree. A special case is finding a smallest spanning tree with a degree constraint; here the time is only O(m + n) more than that needed to find one minimum spanning tree. On transversal and matching matroids, the time is the same as the best-known algorithms for a minimum cost base. This also holds for transversal matroids for convex graphs, which model a scheduling problem on unit-length jobs with release times and deadlines. On partition matroids, a linear-time algorithm is presented. Finally an algorithm related to our general approach finds a smallest spanning tree on a directed graph, where the given root has a degree constraint. Again the time matches the best-known algorithm for the problem without the red element (i.e., degree) constraint.     

Matroid matching in pseudomodular lattices

The matroid matching problem (also known as matroid parity problem) has been intensively studied by several authors. Starting from very special problems, in particular the matching problem and the matroid intersection problem, good characterizations have been obtained for more and more general classes of matroids. The two most recent ones are the class of representable matroids and, later on, the class of algebraic matroids (cf. [4] and [2]). We present a further step of generalization, showing that a good characterization can also be obtained for the class of socalled pseudomodular matroids, introduced by Björner and Lovász (cf. [1]). A small counterexample is included to show that pseudomodularity still does not cover all matroids that behave well with respect to matroid matching.Supported by the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

Regular matroids with every circuit basis fundamental

Hartvigsen and Zemel have obtained a characterization of those graphs which have every circuit basis fundamental. We consider the corresponding problem for binary matroids. We show that, in general, the class of binary matroids for which every circuit basis is fundamental is not closed under the taking of minors. However, this class is closed under the taking of series-minors. We also describe some general properties of this class of matroids. We end by extending Hartvigsen and Zemel's result to the class of regular matroids, thus obtaining a set of excluded minors which are graphic for this class. © 1996 John Wiley & Sons, Inc.     

Oriented hypergraphs: Balanceability

An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camion's algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterization of unbalanceable circuits extends the applications to representable matroids demonstrating that the difference between the Fano and non-Fano matroids is one of balance.     

Submodular linear programs on forests

A general linear programming model for an order-theoretic analysis of both Edmonds' greedy algorithm for matroids and the NW-corner rule for transportation problems with Monge costs is introduced. This approach includes the model of Queyranne, Spieksma and Tardella (1993) as a special case. We solve the problem by optimal greedy algorithms for rooted forests as underlying structures. Other solvable cases are also discussed.     

On some combinatorial properties of algebraic matroids

It was proved implicitly by Ingleton and Main and explicitly by Lindström that if three lines in the algebraic matroid consisting of all elements of an algebraically closed field are not coplanar, but any two of them are, then they pass through one point. This theorem is extended to a more general result about the intersection of subspaces in full algebraic matroids. This result is used to show that the minimax theorem for matroid matching, proved for linear matroids by Lovász, remains valid for algebraic matroids.     

Binary signed-graphic matroids: Representations and recognition algorithms

Recognition algorithms determining whether a given matroid is binary signed-graphic or not are presented in this work. Depending on whether the input is a cographic, a binary or a general matroid different algorithms are provided utilizing mainly decomposition results for the class of signed-graphic matroids. Finally, in order to devise such algorithms, necessary results regarding the representability of signed-graphic matroids in various fields are also given.     

Basis-Transitive Matroids

We consider the problem of classifying all finite basis-transitive matroids and reduce it to the classification of the finite basis-transitive and point-primitive simple matroids (or geometric lattices, or dimensional linear spaces). Our main result shows how a basis- and point-transitive simple matroid is decomposed into a so-called supersum. In particular each block of imprimitivity bears the structure of two closely related simple matroids, and the set of blocks of imprimitivity bears the structure of a point- and basis-transitive matroid.     

Performance-Driven Layer Assignment by Integer Linear Programming and Path-Constrained Hypergraph Partitioning

Performance-driven physical layout design is becoming increasingly important for both high speed integrated circuits and printed circuit boards. This paper studies the problem of assigning wire segments into two layers so as to minimize the number of vias, while taking into account performance constraints such as layer preference and circuit timing. We show that using the Elmore delay model, three timing problems in synchronous digital circuits—the long path problem, the short path problem and the time skew problem—can be formulated as a set of linear inequalities. We use the model of signed hypergraph to represent two-layer routings and formulate the performance-driven optimum layer assignment problem as the path-constrained maximum balance problem in a signed hypergraph. Two solution methods are developed and implemented. First, an integer linear programming formulation is derived for finding exact solutions. Second, a local-search heuristic for hypergraph partitioning is extended to cope with path-inequality constraints. Experimental results on a set of layer-assignment benchmarks demonstrated that the path-constrained local-search heuristic achieves optimum or near-optimum solutions with several orders of magnitude faster than the integer linear programming approach.     

On critical exponents of Dowling matroids

The critical problem in matroid theory is the problem to determine the critical exponent of a given representable matroid over a finite field. In this paper, we study the critical exponents of a class of representable matroids over finite fields, called Dowling matroids. Then the critical problem for a Dowling matroid is corresponding to the classical problem in coding theory to determine the maximum dimension k such that there exists an \([n,k,d]_q\) code for given n, d and q. We give a necessary and sufficient condition on the critical exponents of Dowling matroids by using a coding theoretical approach.     

Branch decomposition heuristics for linear matroids

This paper presents three new heuristics which utilize classification, max-flow, and matroid intersection algorithms respectively to derive near-optimal branch decompositions for linear matroids. In the literature, there are already excellent heuristics for graphs, however, no practical branch decomposition methods for general linear matroids have been addressed yet. Introducing a “measure” which compares the “similarity” of elements of a linear matroid, this work reforms the linear matroid into a similarity graph. Then, the classification method, the max-flow method, and the mat-flow method, all based on the similarity graph, are utilized on the similarity graph to derive separations for a near-optimal branch decomposition. Computational results using the methods on linear matroid instances are shown respectively.     

The Linear Unicyclic Hypergraph with the Second or Third Largest Spectral Radius

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.     

超图的Alcuin数与其横贯数的关系

1000多年前, 英国著名学者Alcuin曾提出过一个古老的渡河问题, 即狼、羊和卷心菜的渡河问题. 最近, Prisner和Csorba等考虑了一般``冲突图"上的渡河问题. 将这一问题推广到超图$H=(V,\mathcal{E})$\,上, 考虑一类情况更一般的运输计划问题. 现在监管者 欲运输超图中的所有点\,(代表``items")\,渡河, 这里$V$的点子 形成超边 当且仅当这些点代表的``items"在无人监管的情况下不能留在一起. 超图$H$的Alcuin数是指超图$H$具有可行运输方案\,(即把$V$的点代表的``items" 全部运到河对岸)\,时船的最小容量. 给出了 $r$-一致完全二部超图和它的伴随超图, 以及$r$-一致超图的Alcuin数, 同时证明了判断$r$-一致超图是否为小船图是NP 困难的.     

Finding lower bounds on the complexity of secret sharing schemes by linear programming

Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants.By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.     

On Steiner trees and minimum spanning trees in hypergraphs

The bottleneck of the state-of-the-art algorithms for geometric Steiner problems is usually the concatenation phase, where the prevailing approach treats the generated full Steiner trees as edges of a hypergraph and uses an LP-relaxation of the minimum spanning tree in hypergraph (MSTH) problem. We study this original and some new equivalent relaxations of this problem and clarify their relations to all classical relaxations of the Steiner problem. In an experimental study, an algorithm of ours which is designed for general graphs turns out to be an efficient alternative to the MSTH approach.     

Combinatorial optimization with interaction costs: Complexity and solvable cases

We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.     

Boolean techniques for matroidal decomposition of independence systems and applications to graphs

The problem of decomposing an independence system into the set-theoretic union of matroids is considered in the first part of this paper and a Boolean procedure is proposed for finding the prime matroidal components of such a decomposition. The second part of the paper deals with the special case of the independence system of all stable sets of a graph, characterizes the graphs whose family of stable sets is the set-theoretic union of two matroids, produces a class of perfect graphs of matroidal number k and gives for graphs an accelerated version of the general decomposition technique.     

On the Hamming distance in combinatorial optimization problems on hypergraph matchings

In this note we consider the properties of the Hamming distance in combinatorial optimization problems on hypergraph matchings, also known as multidimensional assignment problems. It is shown that the Hamming distance between feasible solutions of hypergraph matching problems can be computed as an optimal value of linear assignment problem. For random hypergraph matching problems, an upper bound on the expected Hamming distance to the optimal solution is derived, and an exact expression is obtained in the special case of multidimensional assignment problems with 2 elements in each dimension.     

Oriented matroids

In this paper, the basic properties of oriented matroids are examined. A topological representation theorem for oriented matroids is proven, utilizing the notion of an “arrangement of pseudo-hemispheres”. The duality theorem of linear programming is extended to oriented matroids.