Smallest Cardinality Research Articles

Pricing Equilibria and Graphical Valuations

We characterize pricing equilibria for graphical valuations, which is a class of valuations that admit a compact representation. These valuations are associated with a value graph, whose nodes correspond to items, and edges encode (pairwise) complementarities/substitutabilities between items. It is known that for graphical valuations a Walrasian equilibrium (a pricing equilibrium that relies on anonymous item prices) does not exist in general. Our first contribution is to establish the existence of a pricing equilibrium when the seller uses an agent-specific graphical pricing rule that involves prices for each item and markups/discounts for pairs of items. We then characterize the existence of pricing equilibria with simpler pricing rules which either (i) involve offering prices only for items, or (ii) require anonymity (so that prices are identical for all agents) while allowing for pairwise markups/discounts. We show that a pricing equilibrium with the former pricing rule exists if and only if a Walrasian equilibrium exists, whereas the latter pricing rule may guarantee the existence of a pricing equilibrium even for graphical valuations that do not admit a Walrasian equilibrium. Interestingly, by exploiting a novel connection between the existence of a pricing equilibrium and the partitioning polytope associated with the underlying graph, we also establish that for simple (series-parallel) value graphs a pricing equilibrium with anonymous graphical pricing rule exists if and only if a Walrasian equilibrium exists. These equivalence results imply that simpler pricing rules (i) and (ii) do not guarantee the existence of a pricing equilibrium for all graphical valuations. We extend our results to additively decomposable valuations, which generalize graphical valuations by allowing the valuations to be additive over bundles of cardinality greater than two. We establish that for this class a pricing equilibrium with a pricing rule that similarly decomposes over the same bundles always exists. Our results suggest that for valuations that admit a simple structure (e.g., additively decomposable over bundles of small cardinality), it is always possible to obtain a pricing equilibrium using a simple pricing rule with a similar structure.

On Strong Metric Dimension of Diametrically Vertex Uniform Graphs

A pair of vertices u, v is said to be strongly resolved by a vertex s, if there exist at least one shortest path from s to u passing through v, or a shortest path from s to v passing through u. A set W⊆V, is said to be a strong metric generator if for all pairs u, v ∈/ W, there exist some element s ∈ W such that s strongly resolves the pair u, v. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. The strong met-ric dimension metric dimension problem is to find a min-imum strong metric basis metric basis in the graph.

On the relation between grammatical number and cardinal numbers in development.

This mini-review focuses on the question of how the grammatical number system of a child’s language may help the child learn the meanings of cardinal number words (e.g., “one” and “two”). Evidence from young children learning English, Russian, Japanese, Mandarin, Slovenian, or Saudi Arabic suggests that trajectories of number-word learning differ for children learning different languages. Children learning English, which distinguishes between singular and plural, seem to learn the meaning of the cardinal number “one” earlier than children learning Japanese or Mandarin, which have very little singular/plural marking. Similarly, children whose languages have a singular/dual/plural system (Slovenian and Saudi Arabic) learn the meaning of “two” earlier than English-speaking children. This relation between grammatical and cardinal number may shed light on how humans acquire cardinal-number concepts. There is an ongoing debate about whether mental symbols for small cardinalities (concepts for “oneness,” “twoness,” etc.) are innate or learned. Although an effect of grammatical number on number-word learning does not rule out nativist accounts, it seems more consistent with constructivist accounts, which portray the number-learning process as one that requires significant conceptual change.

ON DYNAMIC MONOPOLIES OF GRAPHS WITH PROBABILISTIC THRESHOLDS

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a graph and ${{\tau }}$ be an assignment of nonnegative thresholds to the vertices of $G$. A subset of vertices, $D$, is an irreversible dynamic monopoly of $(G, \tau )$ if the vertices of $G$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and, for all $i$ with $0 \leq i \leq k-1$, each vertex $v$ in $D_{i+1}$ has at least $\tau (v)$ neighbours in the union of $D_0, D_1, \ldots, D_i$. Dynamic monopolies model the spread of influence or propagation of opinion in social networks, where the graph $G$ represents the underlying network. The smallest cardinality of any dynamic monopoly of $(G,\tau )$ is denoted by $\mathrm{dyn}_{\tau }(G)$. In this paper we assume that the threshold of each vertex $v$ of the network is a random variable $X_v$ such that $0\leq X_v \leq \deg _G(v)+1$. We obtain sharp bounds on the expectation and the concentration of $\mathrm{dyn}_{\tau }(G)$ around its mean value. We also obtain some lower bounds for the size of dynamic monopolies in terms of the order of graph and expectation of the thresholds.

On the strong metric dimension of product graphs

Abstract Let G be a connected graph. A vertex w ∈ V ( G ) strongly resolves two vertices u , v ∈ V ( G ) if there exists some shortest u − w path containing v or some shortest v − w path containing u. A set S of vertices is a strong metric generator for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. In this paper we study the problem of finding sharp bounds for the strong metric dimension of Cartesian product graphs, rooted product graphs and strong product graphs, and express these in terms of invariants of the factor graphs.

New Design of Small Cardinality Model Set for Tracking Controllable-Structure Semiballistic Reentry Vehicle

AbstractThis paper studies maneuver estimation of the controllable-structure semiballistic reentry vehicle (SBRV) by the multiple-model approach. Three kinds of small cardinality model sets are developed to match the vehicle’s uncertain maneuvers: the minimum model set from a new rule for three-dimensional maneuvers, its projection in the mode space’s subspace for two-dimensional maneuvers, and their modifications. Based on these model sets, the multiple-model SBRV tracking estimator is designed to track the vehicle. Simulation results on the vehicle’s random maneuvering demonstrate that the credibility, estimation errors, and sensitivity of the estimators based on the newly designed model set have been improved.

Minimal Zero Norm Solutions of Linear Complementarity Problems

In this paper, we study minimal zero norm solutions of the linear complementarity problems, defined as the solutions with smallest cardinality. Minimal zero norm solutions are often desired in some real applications such as bimatrix game and portfolio selection. We first show the uniqueness of the minimal zero norm solution for Z-matrix linear complementarity problems. To find minimal zero norm solutions is equivalent to solve a difficult zero norm minimization problem with linear complementarity constraints. We then propose a p norm regularized minimization model with p in the open interval from zero to one, and show that it can approximate minimal zero norm solutions very well by sequentially decreasing the regularization parameter. We establish a threshold lower bound for any nonzero entry in its local minimizers, that can be used to identify zero entries precisely in computed solutions. We also consider the choice of regularization parameter to get desired sparsity. Based on the theoretical results, we design a sequential smoothing gradient method to solve the model. Numerical results demonstrate that the sequential smoothing gradient method can effectively solve the regularized model and get minimal zero norm solutions of linear complementarity problems.

Secondary basis unique augmentation matroids and union minimal matroids

By the augmentation axiom of matroids, given two independent sets whose cardinalities are not equal, there exists some other independent set of which the independent set with smaller cardinality is a proper subset. The augmentation axiom requires only the existence of independent set augmentation, not its uniqueness. This causes that the collections of bases of some matroids can be reduced while the unions of the collections of bases of these matroids remain unchanged. We may say these matroids are not minimal. This paper studies a type of matroids whose augmentations of independent sets have some degree of uniqueness and whose collections of bases are minimal in a sense. First, we propose the concept of secondary basis unique augmentation matroid, and prove a matroid is a secondary basis unique augmentation matroid iff the collection of the circuits of its dual matroid is a partition. Then we propose the concept of union minimal matroid based on rank-preserving weak-map, and prove that secondary basis unique augmentation matroids are union minimal matroids. Finally, we propose the concept of basis unique exchange matroid and study its properties.

The product of the restrained domination numbers of a graph and its complement

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted γr(G), is the smallest cardinality of a restrained dominating set of G. In this paper, we show that if G is a graph of order n ≥ 4, then \(\gamma _r \left( G \right)\gamma _r \left( {\bar G} \right) \leqslant 2n\). We also characterize the graphs achieving the upper bound.

An inequality that relates the size of a bipartite graph with its order and restrained domination number

Let $$G=(V,E)$$G=(V,E) be a graph. A set $$S\subseteq V$$S⊆V is a restrained dominating set if every vertex in $$V-S$$V-S is adjacent to a vertex in $$S$$S and to a vertex in $$V-S$$V-S. The restrained domination number of $$G$$G, denoted $$\gamma _{r}(G)$$?r(G), is the smallest cardinality of a restrained dominating set of $$G$$G. Consider a bipartite graph $$G$$G of order $$n\ge 4,$$n?4, and let $$k\in \{2,3,...,n-2\}.$$k?{2,3,...,n-2}. In this paper we will show that if $$\gamma _{r}(G)=k$$?r(G)=k, then $$m\le ((n-k)(n-k+6)+4k-8)/4$$m≤((n-k)(n-k+6)+4k-8)/4. We will also show that this bound is best possible.

Strongly Jónsson and strongly HS modules

Let R be a commutative ring with identity and let M be an infinite unitary R-module. Then M is a Jónsson module provided every proper R-submodule of M has smaller cardinality than M. In this note, we strengthen this condition and call an R-module M (which may be finite) strongly Jónsson provided distinct R-submodules of M have distinct cardinalities. We present a classification of these modules, and then we study a sort of dual notion. Specifically, we consider modules M for which M/N and M/K have distinct cardinalities for distinct R-submodules N and K of M; we call such modules strongly HS (see the introduction for etymology). We conclude the paper with a classification of the strongly HS modules over an arbitrary commutative ring.

A note on multiple coverings of the farthest-off points

Abstract n this work we summarize some recent results, to be included in a forthcoming paper [D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini, and F. Pambianco, Multiple coverigns of the farthest-off points with small density from projective geometry, preprint]. We define μ-density as a characteristic of quality for the kind of coverings codes called multiple coverings of the farthest-off points (MCF). A concept of multiple saturating sets ( ( ρ , μ ) -saturating sets) in projective spaces P G ( N , q ) is introduced. A fundamental relationship of these sets with MCF is showed. Bounds for the smallest possible cardinality of ( 1 , μ ) -saturating sets are obtained. Constructions of small ( 1 , μ ) -saturating sets improving the probabilistic bound are proposed.

The Minimum Consistent Subset Cover Problem: A Minimization View of Data Mining

In this paper, we introduce and study the minimum consistent subset cover (MCSC) problem. Given a finite ground set X and a constraint t, find the minimum number of consistent subsets that cover X, where a subset of X is consistent if it satisfies t. The MCSC problem generalizes the traditional set covering problem and has minimum clique partition (MCP), a dual problem of graph coloring, as an instance. Many common data mining tasks in rule learning, clustering, and pattern mining can be formulated as MCSC instances. In particular, we discuss the minimum rule set (MRS) problem that minimizes model complexity of decision rules, the converse k-clustering problem that minimizes the number of clusters, and the pattern summarization problem that minimizes the number of patterns. For any of these MCSC instances, our proposed generic algorithm CAG can be directly applicable. CAG starts by constructing a maximal optimal partial solution, then performs an example-driven specific-to-general search on a dynamically maintained bipartite assignment graph to simultaneously learn a set of consistent subsets with small cardinality covering the ground set.

Strong tree properties for small cardinals

AbstractAn inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every n ≥ 2 and μ ≥ ℕn, we have (ℕn, μ)-ITP.

Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios

Index tracking is a passive investment strategy in which a fund (e.g., an ETF: exchange traded fund) manager purchases a set of assets to mimic a market index. The tracking error, i.e., the difference between the performances of the index and the portfolio, may be minimized by buying all the assets contained in the index. However, this strategy results in a considerable transaction cost and, accordingly, decreases the return of the constructed portfolio. On the other hand, a portfolio with a small cardinality may result in poor out-of-sample performance. Of interest is, thus, constructing a portfolio with good out-of-sample performance, while keeping the number of assets invested in small (i.e., sparse). In this paper, we develop a tracking portfolio model that addresses the above conflicting requirements by using a combination of L0- and L2-norms. The L2-norm regularizes the overdetermined system to impose smoothness (and hence has better out-of-sample performance), and it shrinks the solution to an equally-weighted dense portfolio. On the other hand, the L0-norm imposes a cardinality constraint that achieves sparsity (and hence a lower transaction cost). We propose a heuristic method for estimating portfolio weights, which combines a greedy search with an analytical formula embedded in it. We demonstrate that the resulting sparse portfolio has good tracking and generalization performance on historic data of weekly and monthly returns on the Nikkei 225 index and its constituent companies.

On The Spectrum of Minimal Defining Sets of Full Designs

A defining set of a t-(v, k, λ) design is a subcollection of the block set of the design which is not contained in any other design with the same parameters. A defining set is said to be minimal if none of its proper subcollections is a defining set. A defining set is said to be smallest if no other defining set has a smaller cardinality. A t-(v, k, λ) design \({D = (V, \fancyscript{B})}\) is called a full design if \({\fancyscript{B}}\) is the collection of all possible k-subsets of V. Every simple t-design is contained in a full design and the intersection of a defining set of a full design with a simple t-design contained in it, gives a defining set of the corresponding t-design. With this motivation, in this paper, we study the full designs when t = 2 and k = 3 and we give several families of non-isomorphic minimal defining sets of full designs. Also, it is proven that there exist values in the spectrum of the full design on v elements such that the number of non-isomorphic minimal defining sets on each of these sizes goes to infinity as v→ ∞. Moreover, the lower bound on the size of the defining sets of the full designs is improved by finding the size of the smallest defining sets of the full designs on eight and nine points. Also, all smallest defining sets of the full designs on eight and nine points are classified.

Hecke-Kiselman monoids of small cardinality

In this paper, we give a characterization of digraphs $Q, |Q|\leq 4$ such that the associated Hecke-Kiselman monoid $H_Q$ is finite. In general, a necessary condition for $H_Q$ to be a finite monoid is that $Q$ is acyclic and its Coxeter components are Dynkin diagram. We show, by constructing examples, that such conditions are not sufficient.

$(\kappa,\theta)$-weak normality

We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.

INSERTION-OF-FACTORS-PROPERTY ON NILPOTENT ELEMENTS

We generalize the insertion-of-factors-property by setting nilpotent products of elements. In the process we introduce the concept of a nil-IFP ring that is also a generalization of an NI ring. It is shown that if K<TEX>$\ddot{o}$</TEX>the's conjecture holds, then every nil-IFP ring is NI. The class of minimal noncommutative nil-IFP rings is completely determined, up to isomorphism, where the minimal means having smallest cardinality.

Perfect sampling of Markov chains with piecewise homogeneous events

Perfect sampling is a technique that uses coupling arguments to provide a sample from the stationary distribution of a Markov chain in a finite time without ever computing the distribution. This technique is very efficient if all the events in the system have monotonicity property. However, in the general (non-monotone) case, this technique needs to consider the whole state space, which limits its application only to chains with a state space of small cardinality. Here, we propose a new approach for the general case that only needs to consider two trajectories. Instead of the original chain, we use two bounding processes (envelopes) and we show that, whenever they meet, one obtains a sample under the stationary distribution of the original chain. We show that this new approach is particularly effective when the state space can be partitioned into pieces where envelopes can be easily computed. We further show that most Markovian queueing networks have this property and we propose efficient algorithms for some of them.