Regular subgraphs of almost regular graphs

AbstractSuppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k ≥ 2q − 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). It is also proved that every simple graph with maximal degree Δ ≥ 2q − 2 and average degree d > ((2q − 2)(2q − 1))(Δ + 1), where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). These results follow from Chevalley's and Olson's theorems on congruences

Helly-Type Theorems in Property Testing

Helly's theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If $S$ is a set of $n$ points in $R^d$, we say that $S$ is $(k,G)$-clusterable if it can be partitioned into $k$ clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object $G$. In this paper, as an application of Helly's theorem, by taking a constant size sample from $S$, we present a testing algorithm for $(k,G)$-clustering, i.e., to distinguish between two cases: when $S$ is $(k,G)$-clusterable, and when it is $\epsilon$-far from being $(k,G)$-clusterable. A set $S$ is $\epsilon$-far $(0<\epsilon\leq1)$ from being $(k,G)$-clusterable if at least $\epsilon n$ points need to be removed from $S$ to make it $(k,G)$-clusterable. We solve this problem for $k=1$ and when $G$ is a symmetric convex object. For $k>1$, we solve a weaker version of this problem. Finally, as an application of our testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability

Combinatorial Alexander Duality -- a Short and Elementary Proof

Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof.Comment: 7 pages, 2 figure; v3: the sign function was simplifie

Hydrostatic pressure effect on Tc of new BiS2 based Bi4O4S3 and NdO0.5F0.5BiS2 layered superconductors

We investigate the external hydrostatic pressure effect on the superconducting transition temperature (Tc) of new layered superconductors Bi4O4S3 and NdO0.5F0.5BiS2. Though the Tc is found to have moderate decrease from 4.8 K to 4.3 K (dTconset/dP = -0.28 K/GPa) for Bi4O4S3 superconductor, the same increases from 4.6 K to 5 K (dTconset/dP = 0.44 K/GPa) upto 1.31 GPa followed by a sudden decrease from 5 K to 4.7 K upto 1.75 GPa for NdO0.5F0.5BiS2 superconductor. The variation of Tc in these systems may be correlated to increase or decrease of the charge carriers in the density of states under externally applied pressure.Comment: 3 pages text +Fig

Role for Interactive Tradespace Exploration in Multi-Stakeholder Negotiations

The significant time, effort, and resource expenditures needed to design and develop aerospace systems motivate on-going research into developing methods for generating, evaluating, and selecting candidate system solutions that can deliver more benefit for a given cost. Compounding the problem is the multiplicity of perspectives of the many stakeholders for such systems, altering the meaning of “benefit” and “cost” depending on the stakeholder considered. Tradespace exploration techniques have been used in the past to generate large datasets in order to gain insights into design-value, cost-benefit tradeoffs for complex aerospace systems. Using interactive tradespace exploration to support multi-stakeholder negotiations can reveal these tradeoffs not only for individuals, but also across a group. A method is introduced and applied to two aerospace cases in order to explore the potential for interactive tradespace exploration to support stakeholder negotiations. Preliminary results indicate the method to be a rapid and beneficial technique, which generated compromise alternatives, guided the elicitation of previously unarticulated information, and resulted in increased confidence and solution buy-in of participating stakeholders.Massachusetts Institute of Technology. Systems Engineering Advancement Research Initiativ

Reconstructing a Simple Polytope from its Graph

Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling order of the facets of the dual polytope of P). Thereby, we derive polynomial certificates for both the vertex-facet incidences as well as for the abstract objective functions in terms of the graph of P. The paper is a variation on joint work with Michael Joswig and Friederike Koerner (2001).Comment: 14 page

Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x : Ax \leq b} along the edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/delta that is a measure for the flatness of the vertices of P. For integer matrices A \in Z^{m \times n} we show a connection between delta and the largest absolute value Delta of any sub-determinant of A, yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This bound is expressed in the same parameter Delta as the recent non-constructive bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(m n^4), which significantly improves the previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer and Frieze

The Least-core and Nucleolus of Path Cooperative Games

Cooperative games provide an appropriate framework for fair and stable profit distribution in multiagent systems. In this paper, we study the algorithmic issues on path cooperative games that arise from the situations where some commodity flows through a network. In these games, a coalition of edges or vertices is successful if it enables a path from the source to the sink in the network, and lose otherwise. Based on dual theory of linear programming and the relationship with flow games, we provide the characterizations on the CS-core, least-core and nucleolus of path cooperative games. Furthermore, we show that the least-core and nucleolus are polynomially solvable for path cooperative games defined on both directed and undirected network

Rigidity and volume preserving deformation on degenerate simplices

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\geq n+1$ if an invariant $c_{k-1}(\alpha^{k-1})$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $c_k(\omega)$ we discovered for any $k$-stress $\omega$ on a cell complex in $M^d$. We introduce a characteristic polynomial of the degenerate simplex by defining $f(x)=\sum_{i=0}^{n+1}(-1)^{i}c_i(\alpha^i)x^{n+1-i}$, and prove that the roots of $f(x)$ are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr