Cardinality Of Subset Research Articles

On the complexity of analysis of automata over a finite ring

A general scheme is investigated that is destined for obtaining estimates based on the cardinality of subsets of a fixed set of automata over some finite commutative-associative ring with unit element. A scheme is proposed to solve parametric systems of polynomial equations based on classes of associated elements of the ring. Some general characteristics of automata over the ring are established.

On connected domination in unit ball graphs

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio approximation algorithm for the minimum connected dominating set problem in unit ball graphs and then introduces and studies the edge-weighted bottleneck connected dominating set problem, which seeks a minimum edge weight in the graph such that the corresponding bottleneck subgraph has a connected dominating set of size k. In wireless network applications this problem can be used to determine an optimal transmission range for a network with a predefined size of the virtual backbone. We show that the problem is hard to approximate within a factor better than 2 in graphs whose edge weights satisfy the triangle inequality and provide a 3-approximation algorithm for such graphs. We also show that for fixed k the problem is polynomially solvable in unit disk and unit ball graphs.

Minimum d-blockers and d-transversals in graphs

We consider a set V of elements and an optimization problem on V: the search for a maximum (or minimum) cardinality subset of V verifying a given property ℘. A d-transversal is a subset of V which intersects any optimum solution in at least d elements while a d-blocker is a subset of V whose removal deteriorates the value of an optimum solution by at least d. We present some general characteristics of these problems, we review some situations which have been studied (matchings, s–t paths and s–t cuts in graphs) and we study d-transversals and d-blockers of stable sets or vertex covers in bipartite and in split graphs.

Minmax regret bottleneck problems with solution-induced interval uncertainty structure

We consider minmax regret bottleneck subset-type combinatorial optimization problems, where feasible solutions are some subsets of a finite ground set of cardinality n . The weights of elements of the ground set are uncertain; for each element, an uncertainty interval that contains its weight is given. In contrast with previously studied interval data minmax regret models, where the set of scenarios (possible realizations of the vector of weights) does not depend on the chosen feasible solution, we consider the problem with solution-induced interval uncertainty structure. That is, for each element of the ground set, a nominal weight from the corresponding uncertainty interval is fixed, and it is assumed that only the weights of the elements included in the chosen feasible solution can deviate from their respective nominal values. This uncertainty structure is motivated, for example, by network design problems, where the weight (construction cost, connection time, etc.) of an edge gets some “real” value, possibly different from its original nominal estimate, only for the edges (connections) that are actually implemented (built); or by capital budgeting problems with uncertain profits of projects, where only the profits of implemented projects can take “real” values different from the original nominal estimates. We present a polynomial O ( n 2 ) algorithm for the problem on a uniform matroid of rank p , where feasible solutions are subsets of cardinality p of the ground set. For the special case where the minimum of the nominal weights is greater than the maximum of the lower-bound weights, we present a simple O ( n + p log p ) algorithm.

New results on optimizing rooted triplets consistency

A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets consistency problem ( MaxRTC ) and the minimum rooted triplets inconsistency problem ( MinRTI ) in which the input is a set R of rooted triplets, and where the objectives are to find a largest cardinality subset of R which is consistent and a smallest cardinality subset of R whose removal from R results in a consistent set, respectively. We first show that a simple modification to Wu’s Best-Pair-Merge-First heuristic Wu (2004) [38] results in a bottom-up-based 3-approximation algorithm for MaxRTC . We then demonstrate how any approximation algorithm for MinRTI could be used to approximate MaxRTC , and thus obtain the first polynomial-time approximation algorithm for MaxRTC with approximation ratio less than 3. Next, we prove that for a set of rooted triplets generated under a uniform random model, the maximum fraction of triplets which can be consistent with any phylogenetic tree is approximately one third. We then provide a deterministic construction of a triplet set having a similar property which is subsequently used to prove that both MaxRTC and MinRTI are NP -hard even if restricted to minimally dense instances. Finally, we prove that unless P = NP , MinRTI cannot be approximated within a ratio of c ⋅ ln n for some constant c > 0 in polynomial time, where n denotes the cardinality of the leaf label set of R .

AN IMPROVED LINE-SEPARABLE ALGORITHM FOR DISCRETE UNIT DISK COVER

Given a set [Formula: see text] of m unit disks and a set [Formula: see text] of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset [Formula: see text] to cover [Formula: see text]. This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by Carmi et al. [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n( log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of Carmi et al. [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem.

Linear and quadratic programming approaches for the general graph partitioning problem

The graph partitioning problem is to partition the vertex set of a graph into a number of nonempty subsets so that the total weight of edges connecting distinct subsets is minimized. Previous research requires the input of cardinalities of subsets or the number of subsets for equipartition. In this paper, the problem is formulated as a zero-one quadratic programming problem without the input of cardinalities. We also present three equivalent zero-one linear integer programming reformulations. Because of its importance in data biclustering, the bipartite graph partitioning is also studied. Several new methods to determine the number of subsets and the cardinalities are presented for practical applications. In addition, hierarchy partitioning and partitioning of bipartite graphs without reordering one vertex set, are studied.

On Relatively Prime Subsets and Supersets

AbstractA nonempty finite set of positive integers

Sets with integral distances in finite fields

Given a positive integer n n , a finite field F q \mathbb F_q of q q elements ( q q odd), and a non-degenerate quadratic form Q Q on F q n \mathbb {F}_q^n , in this paper we study the largest possible cardinality of subsets E ⊆ F q n \mathcal {E} \subseteq \mathbb {F}_q^n with pairwise integral Q Q -distances; that is, for any two vectors {x} = ( x 1 , … , x n ) , {y} = ( y 1 , … , y n ) ∈ E \textbf {{x}}=(x_1, \ldots ,x_n), \textbf {{y}}=(y_1,\ldots ,y_n) \in \mathcal {E} , one has \[ Q ( {x} − {y} ) = u 2 Q(\textbf {{x}}-\textbf {{y}})=u^2 \] for some u ∈ F q u \in \mathbb F_q .

A Dynamic Programming Algorithm for the Generalized Minimum Filter Placement Problem on Tree Structures

We consider a network where nodes communicate by exchanging information packets whose fields include the address of the sending node and that of the destination node. In the absence of some verification mechanism, an attacking node can send packets to another node using a forged origin address. In this study, we consider an optimization problem of identifying a minimum cardinality subset of verification nodes on a tree such that the number of attacks from any forged origin to any destination is limited to a prescribed level. For the case in which communication is permitted between every node in the tree, we develop an optimal polynomial-time dynamic programming algorithm for this problem. We compare the performance of the dynamic programming algorithm against a mixed-integer programming model on randomly generated tree networks at varied levels of security.

Polynomial flow-cut gaps and hardness of directed cut problems

We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given ann-vertex graphGalong withksource-sink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all source-sink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of source-sink pairs that are separated by this deletion. The natural linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, while the standard LP-relaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also theflow-cut gap: the largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem.Our first result is that the flow-cut gap between maximum multicommodity flow and minimum multicut is Ω˜(n1/7) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a long-standing lower bound of Ω(logn) for both types of flow-cut gaps. We notice that these polynomially large flow-cut gaps are in a sharp contrast to the undirected setting where both these flow-cut gaps are known to be Θ(logn). Our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2Ω(log1-ϵn)for any constant ϵ > 0, unless NP ⊆ ZPP. This improves upon the recent Ω(logn/log logn)-hardness result for these problems. We also show that existence of PCP's for NP with perfect completeness, polynomially small soundness, and constant number of queries would imply a polynomial factor hardness of approximation for both these problems. All our results hold for directed acyclic graphs.

On two open problems of 2-interval patterns

The 2-interval pattern problem, introduced in [Stéphane Vialette, On the computational complexity of 2-interval pattern matching problems Theoret. Comput. Sci. 312 (2–3) (2004) 223–249], models general problems with biological structures such as protein contact maps and macroscopic describers of secondary structures of ribonucleic acids. Given a set of 2-intervals D and a model R , the problem is to find a maximum cardinality subset D ′ of D such that any two 2-intervals in D ′ satisfy R , where R is a subset of relations on disjoint 2-intervals: precedence ( < ) , nest ( ⊏ ) , and cross ( ≬ ) . The problem left unanswered at present is that of whether there is a polynomial time solution for the 2-interval pattern problem, when R = { < , ≬ } and all the support intervals of D are disjoint. In this paper, we present a reduction from the clique problem to show that, in this case, the problem is NP-hard. The disjoint 2-interval pattern matching problem is to decide whether a disjoint 2-interval pattern (called the pattern) is a substructure of another disjoint 2-interval pattern (called the target). In general, the problem is NP-hard, but when there are restrictions on the form of the pattern, the problem can, in some cases, be solved in polynomial time. In particular, a polynomial time algorithm has been proposed (Gramm, WABI 2004 and IEEE/ACM TCBB 2004) for the case where the patterns are so-called crossing contact maps. In this paper we show that the problem is actually NP-hard and point out an error in the analysis of the above algorithm. 1 1 The second part of this paper appeared in WABI’2006.

Approximating the 2-interval pattern problem

We address the issue of approximating the 2-Interval Pattern problem over its various models and restrictions. This problem, motivated by RNA secondary structure prediction, asks to find a maximum cardinality subset of a 2-interval set with respect to some prespecified geometric constraints. We present several constant factor approximation algorithms whose performance guarantee depends on the different possible restrictions imposed on the input 2-interval set. In addition, we show that our results extend to the weighted variant of the problem.

An algorithm for extracting maximum cardinality subsets with perfect dominance or anti‐Robinson structures

A common criterion for seriation of asymmetric matrices is the maximization of the dominance index, which sums the elements above the main diagonal of a reordered matrix. Similarly, a popular seriation criterion for symmetric matrices is the maximization of an anti-Robinson gradient index, which is associated with the patterning of elements in the rows and columns of a reordered matrix. Although perfect dominance and perfect anti-Robinson structure are rarely achievable for empirical matrices, we can often identify a sizable subset of objects for which a perfect structure is realized. We present and demonstrate an algorithm for obtaining a maximum cardinality (i.e. the largest number of objects) subset of objects such that the seriation of the proximity matrix corresponding to the subset will have perfect structure. MATLAB implementations of the algorithm are available for dominance, anti-Robinson and strongly anti-Robinson structures.

Extracting constrained 2-interval subsets in 2-interval sets

2-interval sets were used in [S. Vialette, Pattern matching over 2-intervals sets, in: Proc. 13th Annual Symposium Combinatorial Pattern Matching, CPM 2002, in: Lecture Notes in Computer Science, vol. 2373, Springer-Verlag, 2002, pp. 53–63; S. Vialette, On the computational complexity of 2-interval pattern matching, Theoret. Comput. Sci. 312 (2–3) (2004) 223–249] for establishing a general representation for macroscopic describers of RNA secondary structures. In this context, we have a 2-interval for each legal local fold in a given RNA sequence, and a constrained pattern made of disjoint 2-intervals represents a putative RNA secondary structure. We focus here on the problem of extracting a constrained pattern in a set of 2-intervals. More precisely, given a set of 2-intervals D and a model R describing if two disjoint 2-intervals in a solution can be in precedence order (<), be allowed to nest ( ⊏ ) and/or be allowed to cross ( ≬ ), we consider the problem of finding a maximum cardinality subset D ′ ⊆ D of disjoint 2-intervals such that any two 2-intervals in D ′ agree with R . The different combinations of restrictions on model R alter the computational complexity of the problem, and need to be examined separately. In this paper, we improve the time complexity of [S. Vialette, On the computational complexity of 2-interval pattern matching, Theoret. Comput. Sci. 312 (2–3) (2004) 223–249] for model R = { ⊏ } by giving an optimal O ( n log n ) time algorithm, where n is the cardinality of the 2-interval set D . We also give a graph-like relaxation for model R = { ⊏ , ≬ } that is solvable in O ( n 2 n ) time. Finally, we prove that the considered problem is NP-complete for model R = { < , ≬ } even for same-length intervals, and give a fixed-parameter tractability result based on the crossing structure of D .

Feedback vertex sets in mesh-based networks

In this paper, we consider the minimum feedback vertex set problem in graphs, i.e., the problem of finding a minimal cardinality subset of the vertices, whose removal makes a graph acyclic. The problem is N P -hard for general topologies, but optimal and near-optimal solutions have been provided for particular networks. In this paper, the problem is considered for undirected graphs with the following topologies: two- and higher-dimensional meshes of trees, trees of meshes, and pyramid networks. For the two-dimensional meshes of trees the results are optimal; for the higher-dimensional meshes of trees and the tree of meshes the results are asymptotically optimal. For the pyramid networks, there remains a small factor between the upper and the lower bounds.

Some extensions of the Cauchy-Davenport theorem

Abstract The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in Z / p Z , the cardinality of the sumset A + B = { a + b | a ∈ A , b ∈ B } is bounded below by min ( r + s − 1 , p ) ; moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers r , s ⩽ | G | , the analogous sharp lower bound, namely the function μ G ( r , s ) = min { | A + B | | A , B ⊂ G , | A | = r , | B | = s } . Important progress on this topic has been achieved in recent years, leading to the determination of μ G for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.

Call Control in Rings

The call control problem is an important optimization problem encountered in the design and operation of communication networks. The goal of the call control problem in rings is to compute, for a given ring network with edge capacities and a set of paths in the ring, a maximum cardinality subset of the paths such that no edge capacity is violated. We give a polynomial-time algorithm to solve the problem optimally. The algorithm is based on a decision procedure that checks whether a solution with at least k paths exists, which is in turn implemented by an iterative greedy approach operating in rounds. We show that the algorithm can be implemented efficiently and, as a by-product, obtain a linear-time algorithm to solve the problem in chains optimally. For the weighted version of call control in rings, where each path is associated with a weight and the goal is to maximize the total weight of the paths in the solution, we present a simple 2-approximation algorithm and a polynomial-time approximation scheme. While the complexity of the weighted version remains open, we show that it is at least as hard as the bipartite exact matching problem, which has not been resolved to be in P or NP-hard. This latter result follows from recent work by Hochbaum and Levin.

Equi-partitioning of Higher-dimensional Hyper-rectangular Grid Graphs

A d-dimensional grid graph G is the graph on a finite subset in the integer lattice Z d in which a vertex x = (x1, x2, � � � , xn) is joined to another vertex y = (y1, y2, � � � , yn) if for some i we have |xi − yi| = 1 and xj = yj for all j 6 i. G is hyper-rectangular if its set of vertices forms [K1] × [K2] × � � � × [Kd], where each Ki is a nonnegative integer, [Ki] = {0,1, � � � , Ki −1}. The surface area of G is the number of edges between G and its complement in the integer grid Z d . We consider the Minimum Surface Area problem, MSA(G, V ), of partitioning G into subsets of cardinality V so that the total surface area of the subgraphs corresponding to these subsets is a minimum. We present an equi-partitioning algorithm for higher dimensional hyper-rectangles and establish related asymptotic optimality properties. Our algorithm generalizes the two dimensional algorithm due to Martin [8]. It runs in linear time in the number of nodes (O(n), n = |G|) when each Ki is O(n 1/d ). Utilizing a result due to Bollabas and Leader [3], we derive a useful lower bound for the surface area of an equi-partition. Our computational results either achieve this lower bound (i.e., are optimal) or stay within a few percent of the bound.

Exact and heuristic algorithms for solving the generalized minimum filter placement problem

We consider a problem of placing route-based filters in a communication network to limit the number of forged address attacks to a prescribed level. Nodes in the network communicate by exchanging packets along arcs, and the originating node embeds the origin and destination addresses within each packet that it sends. In the absence of a validation mechanism, one node can send packets to another node using a forged origin address to launch an attack against that node. Route-based filters can be established at various nodes on the communication network to protect against these attacks. A route-based filter examines each packet arriving at a node, and determines whether or not the origin address could be legitimate, based on the arc on which the packet arrives, the routing information, and possibly the destination. The problem we consider seeks to find a minimum cardinality subset of nodes to filter so that the prescribed level of security is achieved. We formulate a mixed-integer programming model for the problem and derive valid inequalities for this model by identifying polynomially-solvable subgraphs of the communication network. We also present three heuristics for solving the filter placement problem and evaluate their performance against the optimal solution provided by the mixed-integer programming model.