Showing papers on "K-tree published in 2016"
01 Apr 2016
TL;DR: This paper designs an algorithm to maintain k candidates in the process of maximal clique enumeration, which has limited memory footprint and can achieve a guaranteed approximation ratio, and introduces a novel light-weight $$\mathsf {PNP}$$PNP-$$\ mathsf {Index}$$Index, based on which an optimal maximal cliques maintenance algorithm is designed.
Abstract: Maximal clique enumeration is a fundamental problem in graph theory and has been extensively studied. However, maximal clique enumeration is time-consuming in large graphs and always returns enormous cliques with large overlaps. Motivated by this, in this paper, we study the diversified top-k clique search problem which is to find top-k cliques that can cover most number of nodes in the graph. Diversified top-k clique search can be widely used in a lot of applications including community search, motif discovery, and anomaly detection in large graphs. A naive solution for diversified top-k clique search is to keep all maximal cliques in memory and then find k of them that cover most nodes in the graph by using the approximate greedy max k-cover algorithm. However, such a solution is impractical when the graph is large. In this paper, instead of keeping all maximal cliques in memory, we devise an algorithm to maintain k candidates in the process of maximal clique enumeration. Our algorithm has limited memory footprint and can achieve a guaranteed approximation ratio. We also introduce a novel light-weight $$\mathsf {PNP}$$PNP-$$\mathsf {Index}$$Index, based on which we design an optimal maximal clique maintenance algorithm. We further explore three optimization strategies to avoid enumerating all maximal cliques and thus largely reduce the computational cost. Besides, for the massive input graph, we develop an I/O efficient algorithm to tackle the problem when the input graph cannot fit in main memory. We conduct extensive performance studies on real graphs and synthetic graphs. One of the real graphs contains 1.02 billion edges. The results demonstrate the high efficiency and effectiveness of our approach.
82 citations
TL;DR: A distributed algorithm built on a share-nothing architecture for computing the set of maximal cliques is proposed, which effectively address the problem of skewed workload distribution due to high-degree vertices, which also leads to drastically reduced worst-case time complexity for computing maximal clique in common real-world graphs.
Abstract: Maximal cliques are elementary substructures in a graph and instrumental in graph analysis such as the structural analysis of many complex networks, graph clustering and community detection, network hierarchy detection, emerging pattern mining, vertex importance measures, etc. However, the number of maximal cliques is also notoriously large even for many small real world graphs. This size problem gives rise to challenges in both computing and managing the set of maximal cliques. Many algorithms for computing maximal cliques have been proposed in the literature; however, most of them are sequential algorithms that cannot scale due to the high complexity of the problem, while existing parallel algorithms for computing maximal cliques are mostly immature and especially suffer from skewed workload. As for managing the set of maximal cliques, which is essential due to its large size, there is barely any efficient method for querying or updating the set of maximal cliques. In this paper, we first propose a distributed algorithm built on a share-nothing architecture for computing the set of maximal cliques. We effectively address the problem of skewed workload distribution due to high-degree vertices, which also leads to drastically reduced worst-case time complexity for computing maximal cliques in common real-world graphs. Then, we propose a set of fundamental query operations and efficient algorithms to process the queries, to aid more efficient and effective analysis of the set of maximal cliques. Finally, we also devise algorithms to support efficient update maintenance of the set of maximal cliques when the underlying graph is updated. We verify the efficiency of our algorithms for computing, querying, and updating the set of maximal cliques with a range of real-world graphs from different application domains.
28 citations
TL;DR: This article re-examines the pivoting Bron-Kerbosch algorithm for identifying all maximal cliques within a graph and finds that there exist pivot candidates which may be selected with no penalty to the worst-case running time, even if they do not satisfy the established conditions for the known complexity bound.
20 citations
Posted Content•
TL;DR: In this paper, it was shown that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular for a network with n nodes and m arcs.
Abstract: We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with $n$ nodes and $m$ arcs is never more than $m+n-1$. A key step to prove this is to show the same result for classical transportation polytopes.
18 citations
TL;DR: This paper characterize the subclass of strongly chordal graphs that have a clique arrangement without certain bad 2-cycles and show that L, the graph class of k-leaf powers, is contained in that class.
Abstract: In this paper, we use the new notion of clique arrangements to suggest that leaf powers are a natural special case of strongly chordal graphs The clique arrangement $$\mathcal{A}(G)$$A(G) of a chordal graph G is a directed graph that represents the intersections between maximal cliques of G by nodes and the inclusion relation of these vertex subsets by arcs Recently, strongly chordal graphs have been characterized as the graphs that have a clique arrangement without bad k-cycles for $$k \ge 3$$kź3 The class $$\mathcal{L}_k$$Lk of k-leaf powers consists of graphs $$G=(V,E)$$G=(V,E) that have a k-leaf root, that is, a tree T with leaf set V, where $$xy \in E$$xyźE if and only if the T-distance between x and y is at most k Structural characterizations and linear time recognition algorithms have been found for 2-, 3-, 4-, and, to some extent, 5-leaf powers, and it is known that the union of all k-leaf powers, that is, the graph class $$\mathcal{L}= \bigcup _{k=2}^\infty \mathcal{L}_k$$L=źk=2źLk, forms a proper subclass of strongly chordal graphs Despite that, no essential progress has been made lately In this paper, we characterize the subclass of strongly chordal graphs that have a clique arrangement without certain bad 2-cycles and show that $$\mathcal{L}$$L is contained in that class
16 citations
TL;DR: This paper proposes a new algorithm, called GP, for maximal clique enumeration, and presents a hybrid algorithm, which can effectively leverage the advantages of both the GP algorithm and the classical Bron-and-Kerbosch (BK) algorithm.
Abstract: In a wide variety of emerging data-intensive applications, such as social network analysis, Web document clustering, entity resolution, and detection of consistently co-expressed genes in systems biology, the detection of dense subgraphs (cliques) is an essential component.
Unfortunately, this problem is NP-Complete and thus computationally intensive at scale—hence there is a need for efficient processing, as well as the techniques for distributing the computation across multiple machines such that the computation, which is too time-consuming on a single machine, can be efficiently performed on a machine cluster given that it is large enough. In this paper, we propose a new algorithm (called GP) for maximal clique enumeration.
It identifies cliques by the operation of binary graph partitioning, which iteratively divides a graph until each task is sufficiently small to be processed in parallel. Given a connected graph $$G=(V,E)$$
, the GP algorithm has a space complexity of O(|E|) and a time complexity of $$O(|E|\mu (G))$$
, where $$\mu (G)$$
represents the number of different cliques existing in G. We also present a hybrid algorithm, which can effectively leverage the advantages of both the GP algorithm and the classical Bron-and-Kerbosch (BK) algorithm. Then, we develop corresponding parallel solutions based on the GP and hybrid algorithms. Finally, we evaluate the performance of the proposed solutions on real and synthetic graph data. Our extensive experiments show that in both centralized and parallel setting, our proposed GP and hybrid approaches achieve considerably better performance than the state-of-the-art BK approach. Our parallel solutions are implemented and evaluated on MapReduce, a popular shared-nothing parallel framework, but can easily generalize to other shared-nothing or shared-memory parallel frameworks.
16 citations
TL;DR: It is proved that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if there are no six 4-cliques sharing a 3-clique.
Abstract: We investigate in this paper the relation between Apollonian d-ball packings and stacked $$(d+1)$$(d+1)-polytopes for dimension $$d\ge 3$$d?3. For $$d=3$$d=3, the relation is fully described: we prove that the 1-skeleton of a stacked 4-polytope is the tangency graph of an Apollonian 3-ball packing if and only if there is no six 4-cliques sharing a 3-clique. For higher dimension, we have some partial results.
15 citations
TL;DR: It is proved that the local clique cover number of every claw-free graph is at most cΔ/logΔ, where Δ is the maximum degree of the graph and c is a constant.
Abstract: A k-clique covering of a simple graph G is a collection of cliques of G covering all the edges of G such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called the local clique cover number of G and is denoted by lccG. Local clique cover number can be viewed as the local counterpart of the clique cover number that is equal to the minimum total number of cliques covering all edges. In this article, several aspects of the local clique covering problem are studied and its relationships to other well-known problems are discussed. In particular, it is proved that the local clique cover number of every claw-free graph is at most cΔ/logΔ, where Δ is the maximum degree of the graph and c is a constant. It is also shown that the bound is tight, up to a constant factor. Moreover, regarding a conjecture by Chen eti¾?al. Clique covering the edges of a locally cobipartite graph, Discrete Math 2191-32000, 17-26, we prove that the clique cover number of every connected claw-free graph on n vertices with the minimum degree i¾?, is at most n+ci¾?4/3log1/3i¾?, where c is a constant.
13 citations
TL;DR: It is obtained, for the first time, the complete list of d-dimensional 2-level polytopes up to affine equivalence for dimension d = 7, and geometric constructions that the authors call suspensions play a prominent role in both theoretical and experimental results.
Abstract: A (convex) polytope is said to be 2-level if for every facet-defining direction of hyperplanes, its vertices can be covered with two hyperplanes of that direction. These polytopes are motivated by questions, e.g., in combinatorial optimization and communication complexity. We study 2-level polytopes with one prescribed facet. Based on new general findings about the structure of 2-level polytopes, we give a complete characterization of the 2-level polytopes with some facet isomorphic to a sequentially Hanner polytope, and improve the enumeration algorithm of Bohn et al. (ESA 2015). We obtain, for the first time, the complete list of d-dimensional 2-level polytopes up to affine equivalence for dimension \(d = 7\). As it turns out, geometric constructions that we call suspensions play a prominent role in both our theoretical and experimental results. This yields exciting new research questions on 2-level polytopes, which we state in the paper.
9 citations
Posted Content•
TL;DR: This work considers the maintenance of the set of all maximal cliques in a dynamic graph that is changing through the addition or deletion of edges, and presents nearly tight bounds on the magnitude of change, as well as the first change-sensitive algorithms for clique maintenance, whose runtime is proportional to the magnitude.
Abstract: We consider the maintenance of the set of all maximal cliques in a dynamic graph that is changing through the addition or deletion of edges. We present nearly tight bounds on the magnitude of change in the set of maximal cliques, as well as the first change-sensitive algorithms for clique maintenance, whose runtime is proportional to the magnitude of the change in the set of maximal cliques. We present experimental results showing that these algorithms are efficient in practice.
8 citations
TL;DR: The values of thedeficiency of all k -trees that have maximum degree bounded from above by 2 k are established and it is proved that for each n -vertex graph G of odd order theinequality d e f ( G ) holds.
01 Jan 2016
TL;DR: The paper presents the results of correlation analysis between node centrality and the maximal clique size that each node is part of in complex real-world network graphs, ranging from regular random graphs to scale-free graphs.
Abstract: The paper presents the results of correlation analysis between node centrality (a computationally lightweight metric) and the maximal clique size (a computationally hard metric) that each node is part of in complex real-world network graphs, ranging from regular random graphs to scale-free graphs. The maximal clique size for a node is the size of the largest clique (number of constituent nodes) the node is part of. The correlation coefficient between the centrality value and the maximal clique size for a node is observed to increase with increase in the spectral radius ratio for node degree (a measure of the variation of node degree in the network). As the real-world networks get increasingly scale-free, the correlation between the centrality value and the maximal clique size increases. The degree-based centrality metrics are observed to be relatively better correlated with the maximal clique size compared to the shortest path-based centrality metrics.
TL;DR: A general scheme for constructing polytopes is implemented in this article specifically for the classes of the most important 3D polytes, namely those whose vertices are labeled by integers relative to a particular basis, here called the ω-basis.
Abstract: A general scheme for constructing polytopes is implemented here specifically for the classes of the most important 3D polytopes, namely those whose vertices are labeled by integers relative to a particular basis, here called the ω-basis. The actual number of non-isomorphic polytopes of the same group has no limit. To put practical bounds on the number of polytopes to consider for each group we limit our consideration to polytopes with dominant point (vertex) that contains only nonnegative integers in ω-basis. A natural place to start the consideration of polytopes from is the generic dominant weight which were all three coordinates are the lowest positive integer numbers. Contraction is a continuous change of one or several coordinates to zero.
16 May 2016
TL;DR: 2-level polytopes naturally appear in several areas of mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics.
Abstract: 2-level polytopes naturally appear in several areas of mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics.
Posted Content•
TL;DR: It is proved that the minimum degree of Pedigree polytopes is asymptotically equal to the number of vertices, i.e., the graph is "asymptotical almost complete".
Abstract: Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes whose graphs (1-skeletons) contain the TSP polytope graphs as spanning subgraphs. While deciding adjacency of vertices in TSP polytopes is coNP-complete, Arthanari has given a combinatorial (polynomially decidable) characterization of adjacency in Pedigree polytopes. Based on this characterization, we study the graphs of Pedigree polytopes asymptotically, for large numbers of cities. Unlike TSP polytope graphs, which are vertex transitive, Pedigree graphs are not even regular. Using an "adjacency game" to handle Arthanari's intricate inductive characterization of adjacency, we prove that the minimum degree is asymptotically equal to the number of vertices, i.e., the graph is "asymptotically almost complete".
IBM1
TL;DR: Solving a long-standing open question, NP-hardness is shown, even if the bound on the clique sizes is constant, and a PTAS is given for this case.
01 Jan 2016
Abstract: Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distancebased topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index(PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well.
TL;DR: It is conjectured that the zeros that can be deduced by studying the overlaps essentially determine the enumeration formula in this case, and the domains of validity of these polynomials are investigated.
Posted Content•
TL;DR: The main result determines the pathwidth of the product of cliques and provides a lower bound for the search number of the Cartesian product of graphs.
Abstract: In this paper, we provide results for the search number of the Cartesian product of graphs. We consider graphs on opposing ends of the spectrum: paths and cliques. Our main result determines the pathwidth of the product of cliques and provides a lower bound for the search number of the product of cliques. A consequence of this result is a bound for the search number of arbitrary graphs G and H based on their respective clique numbers.
Posted Content•
TL;DR: In this article, it was shown that every n-vertex chordal graph G has a clique transversal of size at most 2(n-1)/7 if n>=5.
Abstract: The following question was raised by Tuza in 1990 and Erdos et al. in 1992: if every edge of an n-vertex chordal graph G is contained in a clique of size at least four, does G have a clique transversal, i.e., a set of vertices meeting all non-trivial maximal cliques, of size at most n/4? We prove that every such graph G has a clique transversal of size at most 2(n-1)/7 if n>=5, which is the best possible bound.
TL;DR: This paper shows the existence of self-clique { 5 , 6 } -graphs whose cliques are all triangles, thus solving the problem posed by Chia and Ong (2012).
Posted Content•
TL;DR: It is shown how algorithm MLS can be modified to compute a pmo (perfect moplex ordering) as well as a clique tree and the minimal separators of a chordal graph and new cliques in the complement graph can be detected by a condition on labels for any labeling structure.
Abstract: Algorithm MLS (Maximal Label Search) is a graph search algorithm which generalizes algorithms MCS, LexBFS, LexDFS and MNS On a chordal graph, MLS computes a peo (perfect elimination ordering) of the graph We show how algorithm MLS can be modified to compute a pmo (perfect moplex ordering) as well as a clique tree and the minimal separators of a chordal graph We give a necessary and sufficient condition on the labeling structure for the beginning of a new clique in the clique tree to be detected by a condition on labels MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure A linear time algorithm computing a pmo and the generators of the maximal cliques and minimal separators wrt this pmo of the complement graph is provided On a non-chordal graph, algorithm MLSM is used to compute an atom tree of the clique minimal separator decomposition of any graph
TL;DR: In this paper, it was shown that for a connected graph G, if for any cut-set, then G has a k-tree, then it is a spanning tree with maximum degree at most k.
Abstract: A k-tree of a connected graph G is a spanning tree with maximum degree
at most k. The rupture degree for a
connected graph G is defined by , where and ,
respectively, denote the order of the largest component and number of
components in . In this
paper, we show that for a connected graph G,
if for any cut-set , then G has a k-tree.
TL;DR: This work solves an open problem posed by Kratochvil and Tuza to determine the complexity of 2-clique-colouring of perfect graphs with all cliques having size at least 3 and determines a hierarchy of nested subclasses of perfect graph classes whereby each graph class is in a distinct complexity class, namely Σ 2 P - complete, NP -complete, and P.
01 Jan 2016
TL;DR: In this article, the hamiltonicity and traceability of k-tree-Halin graphs are investigated, a generalization of Halin graphs, where k is the number of nodes in a tree and C is a cycle.
Abstract: A k -tree-Halin graph is a planar graph \(F\cup C\), where F is a forest with at most k components and C is a cycle, such that V(C) is the set of all leaves of F, C bounds a face and no vertex has degree 2. This is a generalization of Halin graphs. We are investigating here the hamiltonicity and traceability of k-tree-Halin graphs.
TL;DR: If an n -connected graph G satisfies α 2 ( m + 1 ) ( G ) ?