Simplicial cells in arrangements and mutations of oriented matroids

We investigate the combinatorial and topological properties of simplicial cells in arrangements of (pseudo)hyperplanes, using their interpretations in terms of oriented matroids. Simplicial cells have various applications in computational geometry due to the fact that for an arrangement in general position they are in one-to-one correspondence to local changes (mutations) of its combinatorial type. Several characterizations for mutations of oriented matroids, and their relation to geometric realizability questions are being discussed.We give two short proofs for a result of Shannon [30] that every arrangement of n hyperplanes in E ^d has at least n simplicial cells, this bound being sharp for every n and d. The concatenation operation, a construction introduced by Lawrence and Weinberg [21], will be used to generate a large class of representable oriented matroids with this minimal number of mutations.A homotopy theorem is proved, stating that any two arrangements in general position can be transformed into each other be a sequence of representability preserving mutations. Finally, we give an example of an oriented matroid on eight elements with only seven mutations. As a corollary we obtain a new proof for the non-polytopality of the smallest non-polytopal matroid sphere M ⁹ ₉₆₃.Supported, in part, by an Alfred P. Sloane Doctoral Dissertation Fellowship.     

A theorem on the average number of subfaces in arrangements and oriented matroids

It is known that for simple arrangements in thed-dimensional Euclidean spaceR ^d The average number ofj-dimensional subfaces of ak-dimensional face is less than . In this paper, we show that this is also true for all arrangements inR ^d and for all oriented matroids, and we give combinatorial proofs.     

The augmented multiplicative coalescent,bounded size rules and critical dynamics of random graphs

Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of \(n\) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed \(K\ge 1\) , all components of size greater than \(K\) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erd?s–Rényi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from Bhamidi et al. (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime.     

The asymptotic numbers of regular tournaments,Eulerian digraphs and Eulerian oriented graphs

LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn,, for any>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals.     

The existence of 1-factors in line graphs,squares, and total graphs

The line graph L(G) of G is that graph whose vertex set corresponds to the edge set of G such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The square G² of G has the same vertex set as G and two vertices are adjacent if and only if their distance in G is at most two. The total graph T(G) of G is the graph whose vertex set corresponds to the set of vertices and edges of G such that two vertices of T(G) are adjacent if and only if the corresponding elements of G are adjacent or incident. A 1-factor of a graph is a spanning 1-regular subgraph. For a connected graph G, it is shown that a necessary and sufficient condition that L(G) (respectively, G²; respectively, T(G)) have a 1-factor is that L(G) (respectively, G²; respectively, T(G)) have an even number of vertices.     

Ideal weights: Asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation

Sharp inequalities between weight bounds (from the doubling, A_p, and reverse Hölder conditions) and the BMO norm are obtained when the former are near their optimal values. In particular, the BMO norm of the logarithm of a weight is controlled by the square root of the logarithm of its A_∞ bound. These estimates lead to a systematic development of asymptotically sharp higher integrability results for reverse Hölder weights and extend Coifman and Fefferman's formulation of the A_∞ condition as an equivalence relation on doubling measures to the setting in which all bounds become optimal over small scales.     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

The achievable region method in the optimal control of queueing systems; formulations,bounds and policies

We survey a new approach that the author and his co-workers have developed to formulate stochastic control problems (predominantly queueing systems) asmathematical programming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., find linear or nonlinear constraints on the performance vectors that all policies satisfy. We present linear and nonlinear relaxations of the performance space for the following problems: Indexable systems (multiclass single station queues and multiarmed bandit problems), restless bandit problems, polling systems, multiclass queueing and loss networks. These relaxations lead to bounds on the performance of an optimal policy. Using information from the relaxations we construct heuristic nearly optimal policies. The theme in the paper is the thesis that better formulations lead to deeper understanding and better solution methods. Overall the proposed approach for stochastic control problems parallels efforts of the mathematical programming community in the last twenty years to develop sharper formulations (polyhedral combinatorics and more recently nonlinear relaxations) and leads to new insights ranging from a complete characterization and new algorithms for indexable systems to tight lower bounds and nearly optimal algorithms for restless bandit problems, polling systems, multiclass queueing and loss networks.     

Radiocolorings in periodic planar graphs: PSPACE-completeness and efficient approximations for the optimal range of frequencies

The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function such that |Λ(u)−Λ(v)|2, when u,v are neighbors in G, and |Λ(u)−Λ(v)|1 when the distance of u,v in G is two. The discrete number of frequencies used is called order and the range of frequencies used, span. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span (min span RCP) or the order (min order RCP).In this paper, we deal with an interesting, yet not examined until now, variation of the radiocoloring problem: that of satisfying frequency assignment requests which exhibit some periodic behavior. In this case, the interference graph (modelling interference between transmitters) is some (infinite) periodic graph. Infinite periodic graphs usually model finite networks that accept periodic (in time, e.g. daily) requests for frequency assignment. Alternatively, they can model very large networks produced by the repetition of a small graph.A periodic graph G is defined by an infinite two-way sequence of repetitions of the same finite graph G_i(V_i,E_i). The edge set of G is derived by connecting the vertices of each iteration G_i to some of the vertices of the next iteration G_i+1, the same for all G_i. We focus on planar periodic graphs, because in many cases real networks are planar and also because of their independent mathematical interest.We give two basic results:

• We prove that the min span RCP is PSPACE-complete for periodic planar graphs.

• We provide an O(n(Δ(G_i)+σ)) time algorithm (where|V_i|=n, Δ(G_i) is the maximum degree of the graph G_i and σ is the number of edges connecting each G_i to G_i+1), which obtains a radiocoloring of a periodic planar graph G that approximates the minimum span within a ratio which tends to as Δ(G_i)+σ tends to infinity.

We remark that, any approximation algorithm for the min span RCP of a finite planar graph G, that achieves a span of at most αΔ(G)+constant, for any α and where Δ(G) is the maximum degree of G, can be used as a subroutine in our algorithm to produce an approximation for min span RCP of asymptotic ratio α for periodic planar graphs.

The uniqueness of optimal quadrature formula and optimal interpolation knots with multiplicities in L1

The optimal quadrature formula of type (r₁,..., r_n) for the class of w _∞ ^m is unique if 1<2[(r_i+1)/2]<m. And the set of optimal interpolation knots of type (r₁,...,r_n) for W _∞ ^m in the metric of L₁ is existent and unique in case all r_i being even.     

The height of a 4-cycle in triangle-free 1-planar graphs with minimum degree 5

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. It is known that each 1-planar graph has a vertex of degree at most 7, and also either a vertex of degree at most 4 or a cycle of length at most 4. In the article, it is proven that each triangle-free 1-planar graph of degree less than 5 has a 4-cycle that consists of vertices of degree at most 8.     

The measurement of returns to scale under a simultaneous occurrence of multiple solutions in a reference set and a supporting hyperplane

This research theoretically explores the measurement of RTS (Returns to Scale) under a possible occurrence of multiple solutions in DEA (Data Envelopment Analysis). In this study, the occurrence of multiple solutions is classified into Type I and Type II. Type I is an occurrence of multiple solutions in a reference set. Type II is an occurrence of multiple solutions on a supporting hyperplane passing on the reference set. Both Types I and II are very well known among DEA researchers, but previous research has not sufficiently explored a simultaneous occurrence of Type I and Type II in the RTS measurement. The two types of multiple solutions influence a degree of RTS in the DEA measurement. Such a quantitative issue on RTS is examined from the perspective of the Type I and Type II problems. To deal with such difficulties, a new linear programming approach is proposed to identify all efficient DMUs (Decision Making Units) that consist of a reference set, even if multiple solutions occur on the reference set. Based upon the research result, we can identify when multiple solutions of Type I and/or Type II occur on the RTS measurement and how to deal with such difficulties. Our research result makes it possible to measure a degree of scale economies (RTS) under the simultaneous occurrence of Type I and Type II.     

The good, the bad, and the great: Homomorphisms and cores of random graphs

We consider homomorphism properties of a random graph G(n,p), where p is a function of n. A core H is great if for all e∈E(H), there is some homomorphism from H−e to H that is not onto. Great cores arise in the study of uniquely H-colourable graphs, where two inequivalent definitions arise for general cores H. For a large range of p, we prove that with probability tending to 1 as n→∞, G∈G(n,p) is a core that is not great. Further, we give a construction of infinitely many non-great cores where the two definitions of uniquely H-colourable coincide.     

The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or partial product spaces which arise as polyhedral product functors described below. The geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of Goresky-MacPherson concerning complements of certain subspace arrangements, as well as Davis-Januszkiewicz and Buchstaber-Panov concerning moment-angle complexes. Since the stable decompositions here are geometric, they provide corresponding homological decompositions for generalized moment-angle complexes for any homology theory.