Clique Cover Research Articles

Size-constrained graph partitioning polytopes

We consider the problem of clustering a set of items into subsets whose sizes are bounded from above and below. We formulate the problem as a graph partitioning problem and propose an integer programming model for solving it. This formulation generalizes several well-known graph partitioning problems from the literature like the clique partitioning problem, the equi-partition problem and the k -way equi-partition problem. In this paper, we analyze the structure of the corresponding polytope and prove several results concerning the facial structure. Our analysis yields important results for the closely related equi-partition and k -way equi-partition polytopes as well.

The minimum spanning tree problem with conflict constraints and its variations

We consider the minimum spanning tree problem with conflict constraints (MSTC). The problem is known to be strongly NP-hard and computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded whereas the optimization version is still NP-hard. When the conflict graph is a collection of disjoint cliques, (equivalently, when the conflict relation is transitive) we observe that MSTC can be solved in polynomial time. We also identify other special cases of MSTC that can be solved in polynomial time. Exploiting these polynomially solvable special cases we derive strong lower bounds. Also, various heuristic algorithms and feasibility tests are discussed along with preliminary experimental results. As a byproduct of this investigation, we show that if an ϵ -optimal solution to the maximum clique problem can be obtained in polynomial time, then a ( 3 ϵ − 1 ) -optimal solution to the maximum edge clique partitioning (Max-ECP) problem can be obtained in polynomial time. As a consequence, we have a polynomial time approximation algorithm for the Max-ECP with performance ratio O ( n ( log log n ) 2 log 3 n ) , improving the best previously known bound of O ( n ) .

Binary positive semidefinite matrices and associated integer polytopes

We consider the positive semidefinite (psd) matrices with binary entries, along with the corresponding integer polytopes. We begin by establishing some basic properties of these matrices and polytopes. Then, we show that several families of integer polytopes in the literature—the cut, boolean quadric, multicut and clique partitioning polytopes—are faces of binary psd polytopes. Finally, we present some implications of these polyhedral relationships. In particular, we answer an open question in the literature on the max-cut problem, by showing that the rounded psd inequalities define a polytope.

Clique Partitions of Glued Graphs

A glued graph at $K_2$-clone ($K_3$-clone) results from combining twovertex-disjoint graphs by identifying an edge (a triangle) of each originalgraph. The clique covering numbers of these desired glued graphshave been investigated recently. Analogously, weobtain bounds of the clique partition numbers of glued graphs at$K_2$-clones and $K_3$-clones in terms of the clique partition numbers of theiroriginal graphs. Moreover, we characterize gluedgraphs satisfying such bounds.

Loops and multiple edges in modularity maximization of networks

The modularity maximization model proposed by Newman and Girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges. However, the applications usually correspond to simple graphs. These graphs are compared to a null model where the degree distribution is maintained but edges are placed at random. Therefore, in this null model there will be loops and possibly multiple edges. Sharp bounds on the expected number of loops, and their impact on the modularity, are derived. Then, building upon the work of Massen and Doye, but using algebra rather than simulation, we propose modified null models associated with graphs without loops but with multiple edges, graphs with loops but without multiple edges and graphs without loops nor multiple edges. We validate our models by using the exact algorithm for clique partitioning of Grötschel and Wakabayashi.

Traffic grooming, routing, and wavelength assignment in an optical WDM mesh networks based on clique partitioning

In wavelength routed optical networks, the number of wavelength channels is limited due to several constraints and each wavelength as well as each lightpath support traffic in the Gbps range. On the other hand, the traffic requested by an individual connection is still in the Mbps range. Therefore, to utilize the network resources (such as bandwidth and transceivers) effectively, several low-speed traffic streams have to be efficiently groomed or multiplexed into one or more high-speed lightpaths. The grooming problem of a static demand is considered as an optimization problem. In this work, we have investigated the traffic grooming problem with the objective of maximizing the network throughput for wavelength-routed mesh networks and map this problem to the clique partitioning problem. We have proposed an algorithm to handle general multi-hop static traffic grooming based on the clique partitioning concept. The efficiency of our approach has been established through extensive simulation on different sets of traffic demands with different bandwidth granularities for different network topologies and compared the approach with existing algorithms.

Clustering with partial information

The Correlation Clustering problem, also known as the Cluster Editing problem, seeks to edit a given graph by adding and deleting edges to obtain a collection of vertex-disjoint cliques, such that the editing cost is minimized. The Edge Clique Partitioning problem seeks to partition the edges of a given graph into edge-disjoint cliques, such that the number of cliques is minimized. Both problems are known to be NP-hard, and they have been previously studied with respect to approximation and fixed-parameter tractability. In this paper we study these two problems in a more general setting that we term fuzzy graphs, where the input graphs may have missing information, meaning that whether or not there is an edge between some pairs of vertices of the input graph can be undecided. For fuzzy graphs the Correlation Clustering and Edge Clique Partitioning problems have previously been studied only with respect to approximation. Here we give parameterized algorithms based on kernelization for both problems. We prove that the Correlation Clustering problem is fixed-parameter tractable on fuzzy graphs when parameterized by ( k , r ) , where k is the editing cost and r is the minimum number of vertices required to cover the undecided edges. In particular we show that it has a polynomial-time reduction to a problem kernel on O ( k 2 + r ) vertices. We provide an analogous result for the Edge Clique Partitioning problem on fuzzy graphs. Using ( k , r ) as parameters, where k bounds the size of the partition, and r is the minimum number of vertices required to cover the undecided edges, we describe a polynomial-time kernelization to a problem kernel on O ( k 4 ⋅ 3 r ) vertices. This implies fixed-parameter tractability for this parameterization. Furthermore we also show that parameterizing only by the number of cliques k , is not enough to obtain fixed-parameter tractability. The problem remains, in fact, NP-hard for each fixed k > 2 .

On the minimum vector rank of multigraphs

The minimum vector rank (mvr) of a graph over a field F is the smallest d for which a faithful vector representation of G exists in F d . For simple graphs, minimum semidefinite rank (msr) and minimum vector rank differ by exactly the number of isolated vertices. We explore the relationship between msr and mvr for multigraphs and show that a result linking the msr of chordal graphs to clique cover number also holds for the mvr of multigraphs. We study the difference between msr and mvr in the removal of duplicate vertices in multigraphs, and relate mvr to certain coloring problems. A subset of vertices and the set of edges among them in G form an induced subgraph of G. We let N(v) denote the neighborhood of a vertex v, which is the set of vertices of G adjacent to v, and set N(v) = {v} ∪ N(v). We also define N1(v) to be the set of vertices of G adjacent to v by exactly one edge, and set N1(v) = {v} ∪ N1(v). Note that, for a simple graph, N1(v) = N(v) for every vertex v. We say a vertex v is singly-isolated if N1(v) is empty. A square matrix M = (mij) is combinatorially symmetric when mij = 0 if and only if mji = 0. If G is a graph on n vertices that has no loops, let C(G, F) denote the set of all n-by-n combinatorially symmetric matrices with entries in the field F such that

A minimum transmission time encoding algorithm in multi-rate wireless networks

Network coding enables router nodes to perform coding operations besides replication and forwarding compared with traditional scheme. The minimum transmission time encoding problem in multi-rate wireless networks was investigated. Firstly, the problem is formulated as a minimum weighted clique partition of graph and proven to be NP-complete. Moreover, a heuristic minimum transmission rate first encoding (MTRFE) algorithm is proposed. The simulation results show that the MTRFE algorithm can decrease up to 16% the transmission time than the minimum number of transmissions encoding algorithm.

Clique‐partitioning connections‐scheduling with faulty switches in dilated benes network

In Dilated Optical Multistage Interconnection Networks constructed by 2 ×2 photonic switches, each input‐to‐output connection can be established along one path through the network. If two connections compete for the same switch, only one connection can be established at a time. For a given set of requested connections with/ without faulty switches, it is desirable to establish them in as few rounds as possible. Optical crosstalk is a problem in Optical Multistage Interconnection Networks and a switch is said to be faulty if it causes excessive crosstalk(s) above a given threshold, whose value can be determined according to certain criteria. In this paper, we study the problem of connections‐scheduling with faulty switches in a dilated re‐arrangeable photonic network, a Dilated Benes Network (DBN). We map the connections‐scheduling problem to a variant of the clique partitioning problem, and then use a heuristic algorithm for scheduling the connections. The simulation results show that the proposed method is efficient and it can maintain the reliability of the DBN network by tolerating some faulty switches.

Is rate adaptation beneficial for inter-session network coding?

This paper considers the interplay between rate adaptation and inter-session network coding gains in wireless mesh or ad hoc networks. Inter-session network coding opportunities at relay nodes depend on packets being overheard by surrounding nodes ? The more packets nodes overhear, the more opportunities relays have to combine packets, resulting in a potential increase in network throughput. Thus, by adapting its transmission rate, a node can increase the range over which its packets are overheard, enabling additional opportunities for coding and increased overall throughput. This paper considers inter-session coding, restricted to a single relay (bottleneck) node, or star network topology. Even for such simple topologies the optimal joint rate adaptation and network coding policy is known to be NP-hard. Optimal rate vector selection is a combinatorial optimization problem, which is NP-hard, and finding optimal coding scheme turns out to be a clique partitioning problem which is also NP-hard. So, we provide heuristics to find a suboptimal rate vector and coding scheme. Additionally, we provide a linear programming formulation for network coding when only pairwise intersession coding is allowed. We evaluate the averaged throughput in two different scenarios, in which relays have different access opportunities, giving some intuition on the impact of rate adaptation in lightly and heavily loaded systems. The gains of joint rate adaptation and network coding are marginal when the relay has a higher access opportunity than other nodes, or when the MAC operates ideally it ranges from 9% to 19% as compared to a network without network coding and 4% over a network using regular network coding. While, when the relay has equal access opportunity as other source nodes, which is more typical of todays MAC protocols under heavy loads, the gain ranges from 40% to 62% as compared to the standard relaying case and is upto around 20% as compared to a network with regular pairwise network coding. This can further be increased to a gain of 40% to 120% by replacing pairwise coding with sub-optimal general network coding scheme.

Clustering Qualitative Data Based on Binary Equivalence Relations: Neighborhood Search Heuristics for the Clique Partitioning Problem

The clique partitioning problem (CPP) requires the establishment of an equivalence relation for the vertices of a graph such that the sum of the edge costs associated with the relation is minimized. The CPP has important applications for the social sciences because it provides a framework for clustering objects measured on a collection of nominal or ordinal attributes. In such instances, the CPP incorporates edge costs obtained from an aggregation of binary equivalence relations among the attributes. We review existing theory and methods for the CPP and propose two versions of a new neighborhood search algorithm for efficient solution. The first version (NS-R) uses a relocation algorithm in the search for improved solutions, whereas the second (NS-TS) uses an embedded tabu search routine. The new algorithms are compared to simulated annealing (SA) and tabu search (TS) algorithms from the CPP literature. Although the heuristics yielded comparable results for some test problems, the neighborhood search algorithms generally yielded the best performances for large and difficult instances of the CPP.

Linear Algorithms for Chordal Graphs of Bounded Directed Vertex Leafage

Abstract The directed vertex leafage of a chordal graph G is the smallest integer k such that G is the intersection graph of subtrees of a rooted directed tree where each subtree has at most k leaves. In this note, we show how to find in time O ( k n ) an optimal colouring, a maximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k if a representation of G is given. In particular, this implies that for any path graph G, the four problems can be solved in time O ( n ) given a path representation of G.

Data reduction and exact algorithms for clique cover

To cover the edges of a graph with a minimum number of cliques is an NP-hard problem with many applications. For this problem we develop efficient and effective polynomial-time data reduction rules that, combined with a search tree algorithm, allow for exact problem solutions in competitive time. This is confirmed by experiments with real-world and synthetic data. Moreover, we prove the fixed-parameter tractability of covering edges by cliques.

A Droplet-Manipulation Method for Achieving High-Throughput in Cross-Referencing-Based Digital Microfluidic Biochips

Digital microfluidic biochips are revolutionizing high-throughput DNA, immunoassays, and clinical diagnostics. As high-throughput bioassays are mapped to digital microfluidic platforms, the need for design automation techniques for pin-constrained biochips is being increasingly felt. However, most prior work on biochips computer-aided design has assumed independent control of the underlying electrodes using a large number of (electrical) input pins. We propose a droplet-manipulation method based on a ldquocross-referencingrdquo addressing method that uses ldquorowrdquo and ldquocolumnsrdquo to access electrodes. By mapping the droplet-movement problem on a cross-referenced chip to the clique-partitioning problem from graph theory, the proposed method allows simultaneous movement of a large number of droplets on a microfluidic array. Concurrency is enhanced through the use of an efficient scheduling algorithm that determines the order in which groups of droplets are moved. The proposed design-automation method facilitates high-throughput applications on a pin-constrained biochip, and it is evaluated using random synthetic benchmarks and a set of multiplexed bioassays.

Modelling Robust Flight-Gate Scheduling as a Clique Partitioning Problem

This paper considers the problem of assigning flights to airport gates. We examine the general case in which an aircraft serving a flight may be assigned to different gates for arrival and departure processing and for optional intermediate parking. Restrictions to this assignment include gate closures and shadow restrictions, i.e., the situation in which certain gate assignments may cause the blocking of neighboring gates. The objectives include maximization of the total assignment preference score, minimization of the number of unassigned flights during overload periods, minimization of the number of tows, as well as maximization of the robustness of the resulting schedule with respect to flight delays. We are presenting a simple transformation of the flight-gate scheduling (FGS) problem to a graph problem, i.e., the clique partitioning problem (CPP). The algorithm used to solve the CPP is a heuristic based on the ejection chain algorithm by Dorndorf and Pesch [Dorndorf, U., E. Pesch. 1994. Fast clustering algorithms. ORSA J. Comput. 6 141–153]. This leads to a very effective approach for solving the original problem.

An approach for cutting large and complex pedigrees for linkage analysis

Utilizing large pedigrees in linkage analysis is a computationally challenging task. The pedigree size limits applicability of the Lander-Green-Kruglyak algorithm for linkage analysis. A common solution is to split large pedigrees into smaller computable subunits. We present a pedigree-splitting method that, within a user supplied bit-size limit, identifies subpedigrees having the maximal number of subjects of interest (eg patients) who share a common ancestor. We compare our method with the maximum clique partitioning method using a large and complex human pedigree consisting of 50 patients with Alzheimer's disease ascertained from genetically isolated Dutch population. We show that under a bit-size limit our method can assign more patients to subpedigrees than the clique partitioning method, particularly when splitting deep pedigrees where the subjects of interest are scattered in recent generations and are relatively distantly related via multiple genealogic connections. Our pedigree-splitting algorithm and associated software can facilitate genome-wide linkage scans searching for rare mutations in large pedigrees coming from genetically isolated populations. The software package PedCut implementing our approach is available at http://mga.bionet.nsc.ru/soft/index.html.

Partition into cliques for cubic graphs: Planar case, complexity and approximation

Given a graph G = ( V , E ) and a positive integer k , the partition into cliques ( pic) decision problem consists of deciding whether there exists a partition of V into k disjoint subsets V 1 , V 2 , … , V k such that the subgraph induced by each part V i is a complete subgraph (clique) of G . In this paper, we establish both the NP-completeness of pic for planar cubic graphs and the Max SNP-hardness of pic for cubic graphs. We present a deterministic polynomial time 5 4 -approximation algorithm for finding clique partitions in maximum degree three graphs.

On the clique partitioning problem in weighted interval graphs

The minimum clique partitioning problem in weighted interval graphs (MCPI) is defined as follows. Given an interval graph with nonnegative node weights, the problem is to partition the nodes into a set of cliques such that the sum of node weights in each clique is no more than a given bound. The objective of the problem is to minimize the number of cliques. Recently, Chen et al. [M. Chen, J. Li, J. Li, W. Li, and L. Wang, Some approximation algorithms for the clique partitioning problem in weighted interval graphs, Theoretical Computer Science 381 (2007), 124–133] proposed three approximation algorithms having constant factors 3, 2.5 and 2, and a linear time optimal algorithm for the case with identical weights. In this paper, we show that their factor 2 algorithm does not achieve the expected approximation ratio and the linear time algorithm cannot give an optimal solution for the identical weights case. We also develop an approximation algorithm with factor 2 for the variable weights case and an exact algorithm for the identical weights case.

A Note on First-Fit Coloring of Interval Graphs

We apply the Column Construction Method (Varadarajan et al., Proceedings of the Fifteenth Annual ACM-SIAM Symposium On Discrete Algorithms, pp. 562–571, 2004) to a minimal clique cover of an interval graph to obtain a new proof that First-Fit is 8-competitive for online coloring interval graphs. This proof also yields a new discovery that in each minimal clique cover of an interval graph G, there is a clique of size \(\frac{\omega(G)}{8}\).