Problem In Bipartite Graphs Research Articles

The algorithmic complexity of bondage and reinforcement problems in bipartite graphs

Let G=(V,E) be a graph. A subset D⊆V is a dominating set if every vertex not in D is adjacent to a vertex in D. The domination number of G, denoted by γ(G), is the smallest cardinality of a dominating set of G. The bondage number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with domination number larger than γ(G). The reinforcement number of G is the smallest number of edges whose addition to G results in a graph with smaller domination number than γ(G). In 2012, Hu and Xu proved that the decision problems for the bondage, the total bondage, the reinforcement and the total reinforcement numbers are all NP-hard in general graphs. In this paper, we improve these results to bipartite graphs.

Theory and Practice in Large Carpooling Problems

We address the carpooling problem as a graph-theoretic problem. If the set of drivers is known in advance, then for any car capacity, the problem is equivalent to the assignment problem in bipartite graphs. Otherwise, when we do not know in advance who will drive their vehicle and who will be a passenger, the problem is NP-hard. We devise and implement quick heuristics for both cases, based on graph algorithms, as well as parallel algorithms based on geometric/algebraic approach. We compare between the algorithms on random graphs, as well as on real, very large, data.

Partition of a Binary Matrix intok(k ≥ 3) Exclusive Row and Column Submatrices Is Difficult

A biclustering problem consists of objects and an attribute vector for each object. Biclustering aims at finding a bicluster—a subset of objects that exhibit similar behavior across a subset of attributes, or vice versa. Biclustering in matrices with binary entries (“0”/“1”) can be simplified into the problem of finding submatrices with entries of “1.” In this paper, we consider a variant of the biclustering problem: thek-submatrix partition of binary matrices problem. The input of the problem contains ann×mmatrix with entries (“0”/“1”) and a constant positive integerk. Thek-submatrix partition of binary matrices problem is to find exactlyksubmatrices with entries of “1” such that theseksubmatrices are pairwise row and column exclusive and each row (column) in the matrix occurs in exactly one of theksubmatrices. We discuss the complexity of thek-submatrix partition of binary matrices problem and show that the problem is NP-hard for anyk≥3by reduction from a biclustering problem in bipartite graphs.

Matching and Weighted [formula omitted]-Packing: Algorithms and Kernels

Parameterized algorithms and kernelization algorithms are presented for the weighted P2-Packing problem, which is a generalization of the famous Graph Matching problem. The parameterized algorithms are based on the following new techniques and observations: (1) new study on structure relationship between graph matchings in general graphs and P2-packings in bipartite graphs; (2) an effective graph bi-partitioning algorithm; and (3) a polynomial-time algorithm for a constrained weighted P2-Packing problem in bipartite graphs. The kernelization algorithms are based on the following new techniques: (1) the application of graph matching in kernelization; (2) a crown reduction structure for weighted problems. These techniques lead to randomized and deterministic parameterized algorithms that significantly improve the previous best upper bounds for the problem for both weighted and unweighted versions. For the kernelization algorithm, by using a weighted version of crown reduction, a kernel of size O(k2) is presented, where k is the given parameter of the problem.

Interval incidence coloring of bipartite graphs

In this paper11The extended abstract of this paper appeared on PPAM 2009, LNCS 6067 (2010) 11–20. we study the problem of interval incidence coloring of bipartite graphs. We show the upper bound for interval incidence coloring number (χii) for bipartite graphs χii≤2Δ, and we prove that χii=2Δ holds for regular bipartite graphs. We solve this problem for subcubic bipartite graphs, i.e. we fully characterize the subcubic graphs that admit 4, 5 or 6 coloring, and we construct a linear time exact algorithm for subcubic bipartite graphs. We also study the problem for bipartite graphs with Δ=4 and we show that 5-coloring is easy and 6-coloring is hard (NP-complete). Moreover, we construct an O(nΔ3.5logΔ) time optimal algorithm for trees.

Analytical models for risk-based intrusion response

Risk analysis has been used to manage the security of systems for several decades. However, its use has been limited to offline risk computation and manual response. In contrast, we use risk computation to drive changes in an operating system’s security configuration. This allows risk management to occur in real time and reduces the window of exposure to attack. We posit that it is possible to protect a system by reducing its functionality temporarily when it is under siege. Our goal is to minimize the tension between security and usability by trading them dynamically. Instead of statically configuring a system, we aim to monitor the risk level, using it to drive the tradeoff between security and utility. The advantage of this approach is that it provides users with the maximum possible functionality for any predefined level of risk tolerance.Risk management can be framed as an exercise in managing the constraints on edge and vertex weights of a tripartite graph, with the partitions corresponding to the threats, vulnerabilities, and assets in the system. If a threat requires a specific permission and affects a particular asset, an edge is added between the threat and the permission that mediates access to the vulnerable resource. Another edge is added between the permission and the asset. The presence of a path from a threat, through a permission check, to an asset contributes an element of risk. Risk can be reduced by denying access to a resource that contains a vulnerability or activating data protection measures. We first show that algorithmic underpinnings of optimal risk management can be formulated as the Partial Vertex Cover (PVC) problem in bipartite graphs. We then experimentally compare several heuristics and a 1+22+∊-approximation algorithm we designed for the problem.

An Ordered Turán Problem for Bipartite Graphs

Let $F$ be a graph. A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way. A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$. A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part. Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles. We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$. In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$. For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.

Cluster-Based Control Channel Allocation in Opportunistic Cognitive Radio Networks

Cognitive radio networks (CRNs) involve extensive exchange of control messages, which are used to coordinate critical network functions such as distributed spectrum sensing, medium access, and routing, to name a few. Typically, control messages are broadcasted on a preassigned common control channel, which can be realized as a separate frequency band in multichannel systems, a given time slot in TDMA systems, or a frequency hopping sequence (or CDMA code) in spread spectrum systems. However, a static control channel allocation is contrary to the opportunistic access paradigm. In this paper, we address the problem of dynamically assigning the control channel in CRNs based on time- and space-varying spectrum opportunities. We propose a cluster-based architecture that allocates different channels for control at various clusters in the network. The clustering problem is formulated as a bipartite graph problem, for which we develop a class of algorithms that provide different tradeoffs between two conflicting factors: number of common channels in a cluster and the cluster size. Clusters are guaranteed to have a desirable number of common channels for control, which facilitates for graceful channel migration when primary radio (PR) activity is detected, without the need for frequent reclustering. We perform extensive simulations that verify the agility of our algorithms in adapting to spatial-temporal variations in spectrum availability.

A proof of Cunninghamʼs conjecture on restricted subgraphs and jump systems

For an undirected graph and a fixed integer k, a 2-matching is said to be k-restricted if it has no cycle of length k or less. The problem of finding a maximum cardinality k-restricted 2-matching is polynomially solvable when k⩽3, and NP-hard when k⩾5. On the other hand, the degree sequences of the k-restricted 2-matchings form a jump system for k⩽3, and do not always form a jump system for k⩾5, which is consistent with the polynomial solvability of the maximization problem. In 2002, Cunningham conjectured that the degree sequences of 4-restricted 2-matchings form a jump system and the maximum cardinality 4-restricted 2-matching can be found in polynomial time.In this paper, we show that the first conjecture is true, that is, the degree sequences of 4-restricted 2-matchings form a jump system. We also show that the maximum weight 4-restricted 2-matchings in a bipartite graph induce an M-concave function on the jump system if and only if the weight function is vertex-induced on every square. This result is also consistent with the polynomial solvability of the maximum weight 4-restricted 2-matching problem in bipartite graphs.

On the NP-completeness of the perfect matching free subgraph problem

Given a bipartite graph G=(U∪V,E) such that ∣U∣=∣V∣ and every edge is labelled true or false or both, the perfect matching free subgraph problem is to determine whether or not there exists a subgraph of G containing, for each node u of U, either all the edges labelled true or all the edges labelled false incident to u, and which does not contain a perfect matching. This problem arises in the structural analysis of differential-algebraic systems. The purpose of this paper is to show that this problem is NP-complete. We show that the problem is equivalent to the stable set problem in a restricted case of tripartite graphs. Then we show that the latter remains NP-complete in that case. We also prove the NP-completeness of the related minimum blocker problem in bipartite graphs with perfect matching.

Localized access point selection in infrastructure wireless LANs with performance guarantee

AbstractEfficient network management can improve the network performance and reduce the cost of system maintenance and administration. One of the important duties assigned to the network management system in a infrastructure wireless LAN (or simply wireless LAN) is to assign users to appropriate accessible access points (APs). The current AP selection schemes in wireless LANs cause an unbalanced load, which in turn reduces the performance of individual users. This has motivated intensive studies attempting to determine efficient methods to balance loads among different APs. Existing works either provide heuristic solutions without performance guarantees or provide centralized solutions which are not desirable for the AP selection problem. In this paper, we study the localized solutions that can provide performance guarantees. We model the AP selection problem as a matching problem in bipartite graph. Our objective is to maximize total load among all APs. We propose a class of localized heuristics based on different user knowledge models. For some of these localized heuristics, we prove that there exists a constant approximation ratio in terms of expected total load through mathematical analysis. Simulations are conducted to verify our results. Copyright © 2009 John Wiley & Sons, Ltd.

Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming

The NP-hard general factor problem asks, given a graph and for each vertex a list of integers, whether the graph has a spanning subgraph where each vertex has a degree that belongs to its assigned list. The problem remains NP-hard even if the given graph is bipartite with partition U⊎V, and each vertex in U is assigned the list {1}; this subproblem appears in the context of constraint programming as the consistency problem for the extended global cardinality constraint. We show that this subproblem is fixed-parameter tractable when parameterized by the size of the second partite set V. More generally, we show that the general factor problem for bipartite graphs, parameterized by |V|, is fixed-parameter tractable as long as all vertices in U are assigned lists of length 1, but becomes $\text {\normalfont W[1]}$ -hard if vertices in U are assigned lists of length at most 2. We establish fixed-parameter tractability by reducing the problem instance to a bounded number of acyclic instances, each of which can be solved in polynomial time by dynamic programming.

The Symbolic OBDD Algorithm for Finding Optimal Semi-matching in Bipartite Graphs

The optimal semi-matching problem is one relaxing form of the maximum cardinality matching problems in bipartite graphs, and finds its applications in load balancing. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate Boolean functions efficiently. OBDD-based symbolic algorithms appear to give improved results for large-scale combinatorial optimization problems by searching nodes and edges implicitly. We present novel symbolic OBDD formulation and algorithm for the optimal semi-matching problem in bipartite graphs. The symbolic algorithm is initialized by heuristic searching initial matching and then iterates through generating residual network, building layered network, backward traversing node-disjoint augmenting paths, and updating semi-matching. It does not require explicit enumeration of the nodes and edges, and therefore can handle many complex executions in each step. Our simulations show that symbolic algorithm has better performance, especially on dense and large graphs.

A Novel Symbolic Algorithm for Maximum Weighted Matching in Bipartite Graphs

The maximum weighted matching problem in bipartite graphs is one of the classic combinatorial optimization problems, and arises in many different applications. Ordered binary decision diagram (OBDD) or algebraic decision diagram (ADD) or variants thereof provides canonical forms to represent and manipulate Boolean functions and pseudo-Boolean functions efficiently. ADD and OBDD-based symbolic algorithms give improved results for large-scale combinatorial optimization problems by searching nodes and edges implicitly. We present novel symbolic ADD formulation and algorithm for maximum weighted matching in bipartite graphs. The symbolic algorithm implements the Hungarian algorithm in the context of ADD and OBDD formulation and manipulations. It begins by setting feasible labelings of nodes and then iterates through a sequence of phases. Each phase is divided into two stages. The first stage is building equality bipartite graphs, and the second one is finding maximum cardinality matching in equality bipartite graph. The second stage iterates through the following steps: greedily searching initial matching, building layered network, backward traversing node-disjoint augmenting paths, updating cardinality matching and building residual network. The symbolic algorithm does not require explicit enumeration of the nodes and edges, and therefore can handle many complex executions in each step. Simulation experiments indicate that symbolic algorithm is competitive with traditional algorithms.

The integral stable allocation problem on graphs

As a generalisation of the stable matching problem Baïou and Balinski (2002) [1] defined the stable allocation problem for bipartite graphs, where both the edges and the vertices may have capacities. They constructed a so-called inductive algorithm, that always finds a stable allocation in strongly polynomial time. Here, we generalise their algorithm for non-bipartite graphs with integral capacities. We show that the algorithm does not remain polynomial, although we also present a scaling technique that makes the algorithm weakly polynomial.

Using underapproximations for sparse nonnegative matrix factorization

Nonnegative matrix factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g., text mining, image processing, microarray data analysis, collaborative filtering, etc. We introduce a novel approach to solve NMF problems, based on the use of an underapproximation technique, and show its effectiveness to obtain sparse solutions. This approach, based on Lagrangian relaxation, allows the resolution of NMF problems in a recursive fashion. We also prove that the underapproximation problem is NP-hard for any fixed factorization rank, using a reduction of the maximum edge biclique problem in bipartite graphs. We test two variants of our underapproximation approach on several standard image datasets and show that they provide sparse part-based representations with low reconstruction error. Our results are comparable and sometimes superior to those obtained by two standard sparse nonnegative matrix factorization techniques.

Approximating Geographical Queries

This article proposes a graph-theoretic methodology for query approximation in Geographic Information Systems, enabling the relaxation of three kinds of query constraints: topological, semantic and structural. An approximate query is associated with a value corresponding to the degree of similarity with the original query. Such a value is computed for topological constraints on the basis of the topological distance between configurations, for semantic constraints using the information content approach, and for structural constraints revisiting the maximum weighted matching problem in bipartite graphs. Finally, the high correlation of our proposal with human judgment is demonstrated by an experiment.

Solving a cut problem in bipartite graphs by linear programming: Application to a forest management problem

Given a bipartite graph G = ( V , E ) , a weight for each node, and a weight for each edge, we consider an extension of the MAX-CUT problem that consists in searching for a partition of V into two subsets V 1 and V 2 such that the sum of the weights of the edges from E that have one endpoint in each set plus the sum of the weights of the nodes from V that are in V 1 , is maximal. We prove that this problem can be modeled as a linear program (with real variables) and therefore efficiently solved by standard algorithms. Then, we show how this result can be applied to model a land allocation problem by a 0–1 linear program. This problem consists in determining the cells of a land area, divided into a matrix of identical square cells, which must be harvested and the cells which must be left in old growth so that the weighted sum of the expected populations of some species is maximized. Some computational results are presented to illustrate the efficiency of the method.

Deriving Compact Test Suites for Telecommunication Software Using Distance Metrics

This paper proposes a string edit distance based test selection method to generate compact test sets for telecommunications software. Following the results of previous research, a trace in a test set is considered to be redundant if its edit distance from others is less than a given parameter. The algorithm first determines the minimum cardinality of the target test set inaccordance with the provided parameter, then it selects the test set with the highest sum of internal edit distances. The selection problem is reduced to an assignment problem in bipartite graphs.

A note on transformations of edge colorings of bipartite graphs

The author and A. Mirumian proved the following theorem: Let G be a bipartite graph with maximum degree Δ and let t , n be integers, t ⩾ n ⩾ Δ . Then it is possible to obtain, from one proper edge t-coloring of G, any proper edge n-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper. In this note we show that if t > Δ then we can transform f to g using only transformations of 2-colored subgraphs. We also correct the algorithm suggested in [A.S. Asratian, Short solution of Kotzig's problem for bipartite graphs, J. Combin. Theory Ser. B 74 (1998) 160–168] for transformation of f to g in the case when t = n = Δ and G is regular.