Cardinality Of Subset Research Articles

Single machine adversarial bilevel scheduling problems

We consider single machine scheduling problems in the context of adversarial bilevel optimization where two agents, the leader and the follower, take decisions on the same jobset and the leader acts first with the aim of inducing the worst possible solution for the follower. Thus, the follower schedules the jobs in order to optimize a given criterion. The considered criteria are the total completion time, the total weighted completion time, the maximum lateness and the number of tardy jobs. We focus on adversarial bilevel scheduling with job selection and data modification. In the case with job selection, the leader selects a fixed cardinality subset of the jobs that the follower schedules next. In the case with data modification, the leader can modify some of the data (processing times, due dates, weights), given a limited budget Q. Thus, the follower schedules the set of jobs with modified data. For all the considered criteria either we provide polynomial-time algorithms or show that they can be solved in the worst-case in pseudo-polynomial time.

Parsimonious sequences of finite sets and their applications to chord progressions and music composition

Parsimony is a broad concept with applications in music theory and composition. Two sets A and B of finite cardinality n (n-sets) are in parsimonious relation if there exists a (n–1)-set C that is included in both A and B. A sequence of n-sets is parsimonious if any two successive sets in the sequence are in parsimonious relation. This work demonstrates that for any set of finite cardinality p, there exist sequences of its n-subsets (n ≤ p) that are circular, non-redundant, exhaustive and parsimonious, and it describes the corresponding algorithm. The image of such a sequence by a bijection and the retrograde sequence keep the same four properties. The consequences of these results for the pitch class (pc) subsets of cardinality n (or n-chords) of a pc set of finite cardinality p (p-tonic scale) are derived and discussed in the context of music harmony, microtonality and composition.

On Sum-Free Subsets of Abelian Groups

In this paper, we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind and is hence not overly technical. We consider answers to questions like the following: How many sum-free subsets are there in a given abelian group G? What are its sum-free subsets of maximum cardinality? What is the maximum cardinality of these sum-free subsets? What does a typical sum-free subset of G look like?

On additive bases in infinite abelian semigroups

Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups and semigroups. We show that, for every infinite abelian group T , the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality k contained in an additive basis of order at most h can be bounded in terms of h and k alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon \mathbf{N} . Also, using invariant means, we address a classical problem, initiated by Erdős and Graham and then generalized by Nash and Nathanson both in the case of \mathbf{N} , of estimating the maximal order X_T(h,k) that a basis of cocardinality k contained in an additive basis of order at most h can have.Among other results, we prove that X_T(h,k)=O(h^{2k+1}) for every integer k \ge 1 . This result is new even in the case where k=1 . Besides the maximal order X_T(h,k) , the typical order S_T(h,k) is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in \mathbf{N} and in abelian groups.

Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$.

An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling

Map labeling is a classical problem in cartography and geographic information systems that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent (i.e., overlap-free) labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each timestep, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sublinear update time. Motivated by this, we study the maximal independent set ( MIS ) and maximum independent set ( Max-IS ) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types: (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS ) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of \( \Omega (\Delta) \) update time (where \( \Delta \) is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes. For uniform rectangles, we first give an algorithm that maintains a 4-approximate Max-IS with \( O(1) \) update time. In a subsequent algorithm, we establish the trade-off between approximation quality \( 2(1+\frac{1}{k}) \) and update time \( O(k^2\log n) \) , for \( k\in \mathbb {N} \) . We conclude with an algorithm that maintains a 2-approximate Max-IS for dynamic sets of unit-height and arbitrary-width rectangles with \( O(\log ^2 n + \omega \log n) \) update time, where \( \omega \) is the maximum size of an independent set of rectangles stabbed by any horizontal line. We implement our algorithms and report the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances. We made several major observations in our empirical study. First, the original approximations are well above their respective worst-case ratios. Second, in comparison with the static approaches, the dynamic approaches show a significant speedup in practice. Third, the approximation algorithms show their predicted relative behavior. The better the solution quality, the worse the update times. Fourth, a simple greedy augmentation to the approximate solutions of the algorithms boost the solution sizes significantly in practice.

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the algorithmic complexity of the following problems: (1) Given a vertex-colored graph X = (V,E,c) , compute a minimum cardinality set of vertices S⊆ V such that no nontrivial automorphism of X fixes all vertices in S . A closely related problem is computing a minimum base S for a permutation group G ≤ Sym( n ) given by generators, i.e., a minimum cardinality subset S ⊆ [ n ] such that no nontrivial permutation in G fixes all elements of S . Our focus is mainly on the parameterized complexity of these problems. We show that when k =| S | is treated as parameter, then both problems are MINI [1]-hard. For the dual problems, where k = n – | S | is the parameter, we give FPT algorithms. (2) A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure.

Technical Note—Masking Anstreicher’s linx Bound for Improved Entropy Bounds

A fundamental NP-hard combinatorial-optimization in the area of statistical designs is the maximum-entropy sampling problem (MESP), which seeks to maximize Shannon's “differential entropy” over all subsets of a prespecified cardinality from a set of n Gaussian random variables. This problem has applications in many areas, such as the redesign of environmental-monitoring networks. Most algorithms for exact solution of MESP are branch-and-bound based, and one of the best upper bounds is based on Anstrecher's recent concave “linx relaxation” of differential entropy. A key paradigm for improving bounds is by “masking” the covariance of the random variables with a correlation matrix. The main result establishes that in the best case, the linx bound can be improved by an amount that is at least linear in n by masking. These and other recent results on the hot topic of MESP are leading to practical algorithms for exact solution of meaningful design problems in applied areas such as environmental statistics.

Some Cardinal and Geometric Properties of the Space of Permutation Degree

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP2(R,τ(A)) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP2(R,τS) contains a closed discrete subset of cardinality c. Moreover, it is shown that the functor SPGn preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SPGnX and SPGnY are also homotopically equivalent. As a result, it has been proved that the functor SPGn is a covariant homotopy functor.

Comments on efficient batch verification test for digital signatures based on elliptic curves

Abstract Batch verification for digital signature scheme is a method to verify multiple signatures simultaneously. The complex exponent test (CE test for short) proposed by Cheon and Lee is one of the most efficient batch verification tests for several digital signature schemes on certain types of elliptic curves (including Koblitz curves). The security of the CE test relies essentially on the cardinality of a subset of a residue ring of the endomorphism ring of an elliptic curve over an ideal. They have evaluated the cardinality of the above subset, and have illustrated the effectiveness of the CE test by using the evaluation. The aim of this paper is to point out that their evaluation contains a flaw. The flaw is generally related to two roots of a quadratic equation which is used in their argument. We mend the flaw of their evaluation. Our correct evaluation shows that the CE test can achieve the same security as the underlying signature scheme on Koblitz curves. As a result, the CE test is a secure batch verification when the underlying signature scheme uses Koblitz curves.

A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve

Let C C be an elliptic curve, w ∈ C w\in C , and let S ⊂ C S\subset C be a finite subset of cardinality at least 3 3 . We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle O C ( w ) \mathcal O_C(w) over ( C , S ) (C,S) which are semistable with respect to a weight vector ( 1 2 , … , 1 2 ) \big (\frac {1}{2}, \dots , \frac {1}{2}\big ) .

New formulations and branch-and-cut procedures for the longest induced path problem

Given an undirected graph G=(V,E), the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset W⊆V such that W induces a simple path in G. In this paper, we propose two new formulations with an exponential number of constraints for the problem, together with effective branch-and-cut procedures for its solution. While the first formulation (cec) is based on constraints that explicitly eliminate cycles, the second one (cut) ensures connectivity via cutset constraints. We compare, both theoretically and experimentally, the newly proposed approaches with a state-of-the-art formulation recently proposed in the literature. More specifically, we show that the polyhedra defined by formulation cut and that of the formulation available in the literature are the same. Besides, we show that these two formulations are stronger in theory than cec. We also propose a new branch-and-cut procedure using the new formulations. Computational experiments show that the newly proposed formulation cec, although less strong from a theoretical point of view, is the best performing approach as it can solve all but one of the 1065 benchmark instances used in the literature within the given time limit. In addition, our newly proposed approaches outperform the state-of-the-art formulation when it comes to the median times to solve the instances to optimality. Furthermore, we perform extended computational experiments considering more challenging and hard-to-solve larger instances and evaluate the impacts on the results when offering initial feasible solutions (warm starts) to the formulations.

Disentangling relationships in symptom networks using matrix permutation methods.

Common outputs of software programs for network estimation include association matrices containing the edge weights between pairs of symptoms and a plot of the symptom network. Although such outputs are useful, it is sometimes difficult to ascertain structural relationships among symptoms from these types of output alone. We propose that matrix permutation provides a simple, yet effective, approach for clarifying the order relationships among the symptoms based on the edge weights of the network. For directed symptom networks, we use a permutation criterion that has classic applications in electrical circuit theory and economics. This criterion can be used to place symptoms that strongly predict other symptoms at the beginning of the ordering, and symptoms that are strongly predicted by other symptoms at the end. For undirected symptom networks, we recommend a permutation criterion that is based on location theory in the field of operations research. When using this criterion, symptoms with many strong ties tend to be placed centrally in the ordering, whereas weakly-tied symptoms are placed at the ends. The permutation optimization problems are solved using dynamic programming. We also make use of branch-search algorithms for extracting maximum cardinality subsets of symptoms that have perfect structure with respect to a selected criterion. Software for implementing the dynamic programming algorithms is available in MATLAB and R. Two networks from the literature are used to demonstrate the matrix permutation algorithms.

Solving maximum quasi-clique problem by a hybrid artificial bee colony approach

Given a graph and a threshold (γ), the maximum quasi-clique (MQC) problem is to find a maximum cardinality subset of vertices in the graph such that the edge density of a corresponding induced subgraph is not less than γ. This problem istended version of the famous maximum clique problem, which arises in a various application domains and is known as NP-hard. In this study, we propose a heuristic method called the hybrid artificial bee colony (HABC) algorithm to address the MQC problem by incorporating several dedicated strategies into the artificial bee colony framework, which starts with an opposition-based initialization phase and then repeatedly alternates between an employed bees phase, an onlooker bees phase and a scout bees phase to perform the search. The employed bees phase and onlooker bees phase employ a decent local search and solution-based tabu search to locate the local optima and intensify the search,whereas the scout bees phase employs an opposition-based mechanism to reconstruct the best-found solutions to escape from the local optima and diversify the search. Experimental results indicate that our proposed HABC algorithm can improve the best-known solutions for 46 out of 112 problem instances and match the best-known solutions for 63 instances. Comparisons with the state-of-the-art heuristics and exact methods demonstrate the merits of the proposed algorithm.

An exact algorithm for a class of geometric set-cover problems

Given a set R of m disjoint finite regions in the 2-dimensional plane, all regions having polygonal boundaries, and given a set D of n discs with fixed centers and radii, we consider the problem of finding a minimum cardinality subset D∗⊆D such that every point in R is covered by at least one disc in D∗. We show that this problem can be solved by using an iterative procedure that alternates between the solution of a traditional set-cover problem and the construction of the Laguerre–Voronoi diagram of a circle set. Computer experiments demonstrate the effectiveness of the proposed algorithm, particularly when the number |D∗| of discs necessary to cover R is low.

The structure of two-valued coalitional strategy-proof social choice functions

The paper investigates the structure of coalitional strategy-proof social choice functions – CSP scfs, for short – whose range is a subset of cardinality two of an arbitrary set A of alternatives. The study is conducted in the case where the voters/agents are allowed to express indifference among elements of A, and the domain of the scfs consists of preference profiles P=(Pv)v∈V over a society V of arbitrary cardinality. A representation formula for the two-valued CSP scfs is obtained that provides the structure of such functions.

Categories of two-colored pair partitions Part II: Categories indexed by semigroups

Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to “half-liberated easy” interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.

A BRKGA-based matheuristic for the maximum quasi-clique problem with an exact local search strategy

Given a graph G = (V, E) and a threshold γ ∈ (0, 1], the maximum cardinality quasi- clique problem consists in finding a maximum cardinality subset C* of the vertices in V such that the density of the graph induced in G by C* is greater than or equal to the threshold γ. This problem has a number of applications in data mining, e.g., in social networks or phone call graphs. We propose a matheuristic for solving the maximum cardinality quasi-clique problem, based on the hybridization of a biased random-key genetic algorithm (BRKGA) with an exact local search strategy. The newly proposed approach is compared with a pure biased random-key genetic algorithm, which was the best heuristic in the literature at the time of writing. Computational results show that the hybrid BRKGA outperforms the pure BRKGA.

On κ-bounded and M-compact reflections of topological spaces

For a topological space X its reflection in a class T of topological spaces is a pair (TX,iX) consisting of a space TX∈T and a continuous map iX:X→TX such that for any continuous map f:X→Y to a space Y∈T there exists a unique continuous map f¯:TX→Y such that f=f¯∘iX. In this paper for an infinite cardinal κ and a nonempty set M of ultrafilters on κ, we study the reflections of topological spaces in the classes Hκ of κ-bounded Hausdorff spaces and HM of M-compact Hausdorff spaces (a topological space X is κ-bounded if the closures of subsets of cardinality ≤κ in X are compact; X is M-compact if any function x:κ→X has a p-limit in X for every ultrafilter p∈M).

STOLARSKY'S INVARIANCE PRINCIPLE FOR FINITE METRIC SPACES

Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general structure behind various forms of the main identity. In this work we consider the case of finite metric spaces, relating the quadratic discrepancy of a subset to a certain function of the distribution of distances in it. Our main results are related to a concrete form of the invariance principle for the Hamming space. We derive several equivalent versions of the expression for the discrepancy of a code, including expansions of the discrepancy and associated kernels in the Krawtchouk basis. Codes that have the smallest possible quadratic discrepancy among all subsets of the same cardinality can be naturally viewed as energy minimizing subsets in the space. Using linear programming, we find several bounds on the minimal discrepancy and give examples of minimizing configurations. In particular, we show that all binary perfect codes have the smallest possible discrepancy.