K-tree

About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.

Any-k: Anytime Top-k Tree Pattern Retrieval in Labeled Graphs

Abstract: Many problems in areas as diverse as recommendation systems, social network analysis, semantic search, and distributed root cause analysis can be modeled as pattern search on labeled graphs (also called "heterogeneous information networks" or HINs). Given a large graph and a query pattern with node and edge label constraints, a fundamental challenge is to nd the top-k matches ac- cording to a ranking function over edge and node weights. For users, it is di cult to select value k . We therefore propose the novel notion of an any-k ranking algorithm: for a given time budget, re- turn as many of the top-ranked results as possible. Then, given additional time, produce the next lower-ranked results quickly as well. It can be stopped anytime, but may have to continues until all results are returned. This paper focuses on acyclic patterns over arbitrary labeled graphs. We are interested in practical algorithms that effectively exploit (1) properties of heterogeneous networks, in particular selective constraints on labels, and (2) that the users often explore only a fraction of the top-ranked results. Our solution, KARPET, carefully integrates aggressive pruning that leverages the acyclic nature of the query, and incremental guided search. It enables us to prove strong non-trivial time and space guarantees, which is generally considered very hard for this type of graph search problem. Through experimental studies we show that KARPET achieves running times in the order of milliseconds for tree patterns on large networks with millions of nodes and edges.

Edge coloring a k-tree into two smaller trees

Abstract: The problem of the edge coloring partial k-tree into two partial p- and q-trees with p, q < k is considered An algorithm is provided to construct such a coloring with p + q = k Usefulness of this result in a Lagrangian decomposition framework to solve certain combinatorial optimization problems is discussed

Enumeration of Perfect Sequences of Chordal Graph

Abstract: A graph is chordal if and only if it has no chordless cycle of length more than three. The set of maximal cliques in a chordal graph admits special tree structures called clique trees. A perfect sequence is a sequence of maximal cliques obtained by using the reverse order of repeatedly removing the leaves of a clique tree. This paper addresses the problem of enumerating all the perfect sequences. Although this problem has statistical applications, no efficient algorithm has been proposed. There are two difficulties with developing this type of algorithms. First, a chordal graph does not generally have a unique clique tree. Second, a perfect sequence can normally be generated by two or more distinct clique trees. Thus it is hard using a straightforward way to generate perfect sequences from each possible clique tree. In this paper, we propose a method to enumerate perfect sequences without constructing clique trees. As a result, we have developed the first polynomial delay algorithm for dealing with this problem. In particular, the time complexity of the algorithm on average is O(1) for each perfect sequence.

Chordal- ( k , ℓ )and strongly chordal- ( k , ℓ )graph sandwich problems

Abstract: In this work, we consider the graph sandwich decision problem for property Π, introduced by Golumbic, Kaplan and Shamir: given two graphs G1=(V,E1) and G2=(V,E2), the question is to know whether there exists a graph G=(V,E) such that E1⊆E⊆E2 and G satisfies property Π. Particurlarly, we are interested in fully classifying the complexity of this problem when we look to the following properties Π: `G is a chordal- (k,l)-graph' and `G is a strongly chordal- (k,l)-graph', for all k,l. In order to do that, we consider each pair of positive values of k and l, exhibiting correspondent polynomial algorithms, or NP-complete reductions. We prove that the strongly chordal- ( k,l ) graph sandwich problem is NP-complete, for k≥1 and l≥1, and that the chordal- ( k,l ) graph sandwich problem is NP-complete, for positive integers k and l such that k+l ≥ 3. Moreover, we prove that both problems are in P when k or l is zero and k+l ≤ 2. To complete the complexity dichotomy concerning these problems for all nonnegative values of k and l, there still remains the open question of settling the complexity for the case k+l ≥ 3 and one of them is equal to zero.

Learning inclusion-optimal chordal graphs

Abstract: Chordal graphs can be used to encode dependency models that are representable by both directed acyclic and undirected graphs. This paper discusses a very simple and efficient algorithm to learn the chordal structure of a probabilistic model from data. The algorithm is a greedy hill-climbing search algorithm that uses the inclusion boundary neighborhood over chordal graphs. In the limit of a large sample size and under appropriate hypotheses on the scoring criterion, we prove that the algorithm will find a structure that is inclusion-optimal when the dependency model of the data-generating distribution can be represented exactly by an undirected graph. The algorithm is evaluated on simulated datasets.