Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of element-disjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k element-connected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns $\lfloor\frac{k}{2} \rfloor-1$ element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching?2.     

Output-Sensitive Reporting of Disjoint Paths

A k -path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper we study the problem of performing k -path queries, with , in a graph G with n vertices. We denote with the total length of the reported paths. For , we present an optimal data structure for G that uses O(n) space and executes k -path queries in output-sensitive time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st ) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs. Received August 24, 1996; revised April 8, 1997.     

New primal–dual algorithms for Steiner tree problems

We present new primal–dual algorithms for several network design problems. The problems considered are the generalized Steiner tree problem (GST), the directed Steiner tree problem (DST), and the set cover problem (SC) which is a subcase of DST. All our problems are NP-hard; so we are interested in their approximation algorithms. First, we give an algorithm for DST which is based on the traditional approach of designing primal–dual approximation algorithms. We show that the approximation factor of the algorithm is k, where k is the number of terminals, in the case when the problem is restricted to quasi-bipartite graphs. We also give pathologically bad examples for the algorithm performance. To overcome the problems exposed by the bad examples, we design a new framework for primal–dual algorithms which can be applied to all of our problems. The main feature of the new approach is that, unlike the traditional primal–dual algorithms, it keeps the dual solution in the interior of the dual feasible region. The new approach allows us to avoid including too many arcs in the solution, and thus achieves a smaller-cost solution. Our computational results show that the interior-point version of the primal–dual most of the time performs better than the original primal–dual method.     

Diversified top-<Emphasis Type="Italic">k</Emphasis> clique search

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 \(\mathsf {PNP}\)-\(\mathsf {Index}\), 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.     

A linear-time algorithm for paired-domination problem in strongly chordal graphs

Let G=(V,E) be a simple graph without isolated vertices. A vertex set S⊆V is a paired-dominating set if every vertex in V−S has at least one neighbor in S and the induced subgraph G[S] has a perfect matching. In this paper, we present a linear-time algorithm to find a minimum paired-dominating set in strongly chordal graphs if the strong (elimination) ordering of the graph is given in advance.     

On the coding of ordered graphs

Ordered graph and ordered graph isomorphism provide a natural representation of many objects in applications such as computational geometry, computer vision and pattern recognition. In the present paper we propose a coding procedure for ordered graphs that improves an earlier one based on Eulerian circuits of graphs in terms of both simplicity and computational efficiency. Using our coding approach, we show that the ordered graph isomorphism problem can be optimally solved in quadratic time, although no efficient (polynomial-bound) isomorphism algorithm for general graphs exists today. An experimental evaluation demonstrates the superior performance of the new method.     

k-tuple domination in graphs

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the k-tuple domination problem in strongly chordal graphs, which is a subclass of chordal graphs and includes trees, block graphs, interval graphs and directed path graphs. We also prove that the k-tuple domination problem is NP-complete for split graphs (a subclass of chordal graphs) and for bipartite graphs.     

On some union and intersection problems for polygons with fixed orientations

Objects with fixed orientations play an important role in many application areas, for instance VLSI design. Problems involving only rectilinearly oriented (rectangular) objects, as a simplest case, have been studied with the VLSI design application in mind. These objects can be transistors, cells or macros. In reality, they are more suitably represented by polygons rather than just rectangles. In this note we describe how to perform a general decomposition of a set of polygons with fixed orientations in order to solve various computational geometry problems which are important in VLSI design. The decomposition is very simple and efficiently computable, and it allows the subsequent application of algorithms for the rectilinear case, leading to some very efficient and some optimal solutions. We illustrate the technique in detail at the problem of finding the connected components of a set of polygons, for which we derive an optimal solution. The wide applicability of the method is then demonstrated at the problem of finding all pairs of intersecting polygons, yielding an optimal solution.The work of this author was partially supported by the National Science Foundation under Grants MCS 8342682 and ECS 8340031. This work was performed while this author was a summer visitor at the IBM T. J. Watson Research Center.     

Finding next-to-shortest paths in a graph

We study the problem of finding the next-to-shortest paths in a graph. A next-to-shortest (u,v)-path is a shortest (u,v)-path amongst (u,v)-paths with length strictly greater than the length of the shortest (u,v)-path. In contrast to the situation in directed graphs, where the problem has been shown to be NP-hard, providing edges of length zero are allowed, we prove the somewhat surprising result that there is a polynomial time algorithm for the undirected version of the problem.     

The K r -Packing Problem

For a fixed integer r≥2, the K _r -packing problem is to find the maximum number of pairwise vertex-disjointK _r 's (complete graphs on r vertices) in a given graph. The K _r -factor problem asks for the existence of a partition of the vertex set of a graph into K _r 's. The K _r -packing problem is a natural generalization of the classical matching problem, but turns out to be much harder for r≥3 – it is known that for r≥3 the K _r -factor problem is NP-complete for graphs with clique number r [16]. This paper considers the complexity of the K _r -packing problem on restricted classes of graphs. We first prove that for r≥3 the K _r -packing problem is NP-complete even when restrict to chordal graphs, planar graphs (for r=3, 4 only), line graphs and total graphs. The hardness result for K ₃-packing on chordal graphs answers an open question raised in [6]. We also give (simple) polynomial algorithms for the K ₃-packing and the K _r -factor problems on split graphs (this is interesting in light of the fact that K _r -packing becomes NP-complete on split graphs for r≥4), and for the K _r -packing problem on cographs. Received September 27, 1999; revised August 14, 2000     

The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs

We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T\mathcal{T} of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to T\mathcal{T} is a set of vertex-disjoint paths ℘ that covers the vertices of G such that the k vertices of T\mathcal{T} are all endpoints of the paths in ℘. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T\mathcal{T} is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49–64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.     

Overlapping community detection in labeled graphs

We present a new approach for the problem of finding overlapping communities in graphs and social networks. Our approach consists of a novel problem definition and three accompanying algorithms. We are particularly interested in graphs that have labels on their vertices, although our methods are also applicable to graphs with no labels. Our goal is to find k communities so that the total edge density over all k communities is maximized. In the case of labeled graphs, we require that each community is succinctly described by a set of labels. This requirement provides a better understanding for the discovered communities. The proposed problem formulation leads to the discovery of vertex-overlapping and dense communities that cover as many graph edges as possible. We capture these properties with a simple objective function, which we solve by adapting efficient approximation algorithms for the generalized maximum-coverage problem and the densest-subgraph problem. Our proposed algorithm is a generic greedy scheme. We experiment with three variants of the scheme, obtained by varying the greedy step of finding a dense subgraph. We validate our algorithms by comparing with other state-of-the-art community-detection methods on a variety of performance measures. Our experiments confirm that our algorithms achieve results of high quality in terms of the reported measures, and are practical in terms of performance.     

Serial and Parallel Algorithms for (k,2)-Partite Graphs

We introduce a class of layered graphs which we call (k,2)-partite and which we argue are an interesting class because of several important applications. We show that testing for (k,2)-partiteness can be done efficiently both on sequential and parallel machines, by showing that membership is in NSPACE(log n) and in NC². We show that (k,2)-partite graphs have bounded path width. We then show that a particular NP-complete problem, namely Maximum Independent Set, is solvable in linear time on bounded pathwidth graphs if the path decomposition is included in the input. Finally, we show that the Maximum Independent Set problem is in NC² for (k,2)-partite graphs. We note that linear time solutions for certain NP-complete problems have been shown for a wider class of graphs, namely partial k-trees. Our linear time algorithm is somewhat simpler in structure. We conjecture that our techniques can be used on many NP-complete problems to yield efficient algorithms for (k,2)-partite graphs.     

Parameterized complexity of the induced subgraph problem in directed graphs

In this Letter, we consider the parameterized complexity of the following problem: Given a hereditary property P on digraphs, an input digraph D and a positive integer k, does D have an induced subdigraph on k vertices with property P? We completely characterize hereditary properties for which this induced subgraph problem is W[1]-complete for two classes of directed graphs: general directed graphs and oriented graphs. We also characterize those properties for which the induced subgraph problem is W[1]-complete for general directed graphs but fixed parameter tractable for oriented graphs. These results are among the very few parameterized complexity results on directed graphs.     

On exact solutions to the Euclidean bottleneck Steiner tree problem

We study the Euclidean bottleneck Steiner tree problem: given a set P of n points in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edge in the tree is minimized. This problem is known to be NP-hard even to approximate within ratio and there was no known exact algorithm even for k=1 prior to this work. In this paper, we focus on finding exact solutions to the problem for a small constant k. Based on geometric properties of optimal location of Steiner points, we present an optimal -time exact algorithm for k=1 and an O(n²)-time algorithm for k=2. Also, we present an optimal -time exact algorithm for any constant k for a special case where there is no edge between Steiner points.     

An experimental evaluation of local search heuristics for graph partitioning

A common problem that arises in many applications is to partition the vertices of a graph intok subsets, each containing a bounded number of vertices, such that the number of graph edges with endpoints in different subsets is minimized. This paper describes an empirical study of the performance of various local search heuristics for thisk-way graph partitioning problem. The heuristics examined are local optimization, simulated annealing, tabu search, and genetic algorithms. In addition, the hierarchical hybrid approach is introduced, in which the problem is recursively decomposed into small pieces, to which local search heuristics are then applied.     

异质信息网络中最大路径连通Steiner分量查询算法

异质信息网络(HINs)是包含多种类型对象(顶点)和链接(边)的有向图,能够表达丰富复杂的语义和结构信息.HINs中的稠密子图查询问题,即给定一个查询点q,在HINs中查询包含q的稠密子图,已成为该领域的热点和重点研究问题,并在活动策划、生物分析和商品推荐等领域具有广泛应用.但现有方法主要存在以下两个问题:(1)基于模体团和关系约束查询的稠密子图具有多种类型顶点,导致其不能解决仅关注某种特定类型顶点的场景;(2)基于元路径的方法虽然可查询到某种特定类型顶点的稠密子图,但其忽略了子图中顶点之间基于元路径的连通度.为此,首先在HINs中提出了基于元路径的边不相交路径的连通度,即路径连通度;然后,基于路径连通度提出了k-路径连通分量(k-PCC)模型,该模型要求子图的路径连通度至少为k;其次,基于k-PCC模型提出了最大路径连通Steiner分量(SMPCC)概念,其为包含q的具有最大路径连通度的k-PCC;最后,提出一种高效的基于图分解的k-PCC发现算法,并在此基础上提出了优化查询SMPCC算法.大量基于真实和合成HINs数据的实验结果验证了所提出模型和算法的有效性和高效性.     

A Tight Lower Bound for a Special Case of Quadratic 0–1 Programming

We consider a still NP-complete partial case of the unconstrained binary quadratic optimization problem that can be described in terms of an undirected graph with red edges having negative weights and green edges having positive weights. The maximum vertex degree of the graph is three. It can be assumed w.l.o.g. that every vertex is incident to a red and a green edge. We are looking for a vertex cover with respect to the red edges which covers a subset of green edges of total weight as small as possible. We prove that for all connected such graphs except a subclass of special graphs having exactly five green edges it is possible to find a vertex cover with respect to the red edges for which the total weight of uncovered green edges is at least 1/4 fraction of the total weight of all green edges.     

Almost Exact Matchings

In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)?1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K _3,3-minor free graphs (these include all planar graphs as well as many others) in O(n ^3.19) worst case time. Our algorithm can also count the number of perfect matchings in K _3,3-minor free graphs in O(n ^2.19) time.