Subgraph Problem Research Articles

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.

Maximum $r$-Regular Induced Subgraph Problem: Fast Exponential Algorithms and Combinatorial Bounds

We show that for a fixed $r$, the number of maximal $r$-regular induced subgraphs in any graph with $n$ vertices is upper bounded by $\mathcal{O}(c^n)$, where $c$ is a positive constant strictly less than $2$. This bound generalizes the well-known result of Moon and Moser, who showed an upper bound of $3^{n/3}$ on the number of maximal independent sets of a graph on $n$ vertices. We complement this upper bound result by obtaining an almost tight lower bound on the number of (possible) maximal $r$-regular induced subgraphs possible in a graph on $n$ vertices. Our upper bound results are algorithmic. That is, we can enumerate all the maximal $r$-regular induced subgraphs in time $\mathcal{O}(c^n n^{\mathcal{O}(1)})$. A related question is that of finding a maximum-sized $r$-regular induced subgraph. Given a graph $G=(V,E)$ on $n$ vertices, the Maximum $r$-Regular Induced Subgraph (M-$r$-RIS) problem asks for a maximum-sized subset of vertices, $R \subseteq V$, such that the induced subgraph on $R$ is $r$-regular. As a by-product of the enumeration algorithm, we get a $\mathcal{O}(c^n)$ time algorithm for this problem for any fixed constant $r$, where $c$ is a positive constant strictly less than $2$. Furthermore, we use the techniques and results obtained in the paper to obtain improved exact algorithms for a special case of the Induced Subgraph Isomorphism problem, namely, the Induced $r$-Regular Subgraph Isomorphism problem, where $r$ is a constant, the $\delta$-Separating Maximum Matching problem and the Efficient Edge Dominating Set problem.

Algorithmic approaches for the minimum rainbow subgraph problem

Abstract We consider the Minimum Rainbow Subgraph problem (MRS): Given a graph G, whose edges are coloured with p colours. Find a subgraph F ⊆ G of minimum order and with p edges such that each colour occurs exactly once. This problem is APX-hard. In this paper we will show that the Greedy algorithm for the MRS problem has an approximation ratio of Δ 2 + ln Δ + 1 2 for graphs with maximum degree Δ. If the average degree d of a minimum rainbow subgraph is known, then the approximation ratio is d 2 + ln ⌈ d ⌉ + 1 2 .

Polyhedral study of the maximum common induced subgraph problem

In this paper we present an exact algorithm for the Maximum Common Induced Subgraph Problem (MCIS) by addressing it directly, using Integer Programming (IP) and polyhedral combinatorics. We study the MCIS polytope and introduce strong valid inequalities, some of which we prove to define facets. Besides, we show an equivalence between our IP model for MCIS and a well-known formulation for the Maximum Clique problem. We report on computational results of branch-and-bound (B&B) and branch-and-cut (B&C) algorithms we implemented and compare them to those yielded by an existing combinatorial algorithm.

A polyhedral study of the maximum edge subgraph problem

The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k, the maximum edge subgraph problem consists of finding a k-vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families. Finally, we implement a branch and cut algorithm for this problem. This computational study illustrates the effectiveness of the classes of inequalities presented in this work.

Cooperative location games based on the minimum diameter spanning Steiner subgraph problem

In this paper we introduce and analyze new classes of cooperative games related to facility location models. The players are the customers (demand points) in the location problem and the characteristic value of a coalition is the cost of serving its members. Specifically, the cost in our games is the service diameter of the coalition.We study the existence of core allocations for these games, focusing on network spaces, i.e., finite metric spaces induced by undirected graphs and positive edge lengths.

Wildlife corridors as a connected subgraph problem

Wildlife corridors connect areas of biological significance to mitigate the negative ecological impacts of habitat fragmentation. In this article we formalize the optimal corridor design as a connected subgraph problem, which maximizes the amount of suitable habitat in a fully connected parcel network linking core habitat areas, subject to a constraint on the funds available for land acquisition. To solve this challenging computational problem, we propose a hybrid approach that combines graph algorithms with Mixed Integer Programming-based optimization. We apply this technique to the design of corridors for grizzly bears in the U.S. Northern Rockies, illustrating the underlying computational complexities by varying the granularity of the parcels available for acquisition. The approach that is introduced is general and can be applied to other species or other similar problems, such as those occurring in social networks.

Combinatorial properties and further facets of maximum edge subgraph polytopes

Abstract Given a graph G and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset of G such that the number of edges within the subset is maximum. This NP-hard problem arises in the analysis of cohesive subgroups in social networks. In this work we study the polytope P ( G , k ) associated with a straightforward integer programming formulation of the maximum edge subgraph problem. We characterize the graph generated by P ( G , k ) and give a tight bound on its diameter. We give a complete description of P ( K 1 n , k ) , where K 1 n is the star on n + 1 vertices, and we conjecture a complete description of P ( m K 2 , k ) , where m K 2 is the graph composed by m disjoint edges. Finally, we introduce three families of facet-inducing inequalities for P ( G , k ) , which generalize known families of valid inequalities for this polytope.

The maximum [formula omitted]-colorable subgraph problem and orbitopes

Given an undirected node-weighted graph and a positive integer k , the maximum k -colorable subgraph problem is to select a k -colorable induced subgraph of largest weight. The natural integer programming formulation for this problem exhibits a high degree of symmetry which arises by permuting the color classes. It is well known that such symmetry has negative effects on the performance of branch-and-cut algorithms. Orbitopes are a polyhedral way to handle such symmetry and were introduced in Kaibel and Pfetsch (2008) [2]. The main goal of this paper is to investigate the polyhedral consequences of combining problem-specific structure with orbitope structure. We first show that the LP-bound of the integer programming formulation mentioned above can only be slightly improved by adding a complete orbitope description. We therefore investigate several classes of facet-defining inequalities for the polytope obtained by taking the convex hull of feasible solutions for the maximum k -colorable subgraph problem that are contained in the orbitope. We study conditions under which facet-defining inequalities for the polytope associated with the maximum k -colorable subgraph problem and the orbitope remain facet-defining for the combined polytope or can be modified to yield facets. It turns out that the results depend on both the structure and the labeling of the underlying graph.

Research on Random Walk Rough Matching Algorithm of Attribute Sub-Graph

In the analysis of social network, the attribute values of an entity change constantly as time goes by and the corresponding attribute graph changes accordingly. However, the essence of the entity is invariable. The problem is how to discover essence from the change of mining frequent rough matching attribute sub-graph in an attribute graph in this paper. The problem of the attribute sub-graph rough matching is defined and described, then the random walk rough matching algorithm of attribute sub-graph is designed so that the problem of incomplete coincide attribute sub-graph rough matching would be solved. Attribute sub-graph is the expansion of the traditional sub-graph; rough matching problem is the extension of traditional sub-graph matching problem. With the help of the random walk rough matching algorithm of attribute sub-graph, more potential frequent attribute sub-graphs can be discovered, and more valuable information mined.

Parameterized complexity of even/odd subgraph problems

We study the parameterized complexity of the problems of determining whether a graph contains a k-edge subgraph ( k-vertex induced subgraph) that is a Π-graph for Π-graphs being one of the following four classes of graphs: Eulerian graphs, even graphs, odd graphs, and connected odd graphs. We also consider the parameterized complexity of their parametric dual problems. For these sixteen problems, we show that eight of them are fixed parameter tractable and four are W[1]-hard. Our main techniques are the color-coding method of Alon, Yuster and Zwick, and the random separation method of Cai, Chan and Chan.

Top-Down Algorithm for Mining Maximal Frequent Subgraph

In order to solution the problem of mining maximal frequent subgraph is very hard we proposed new algorithm Top-Down. The process of this algorithm is using decision tree to count support then firstly judge the biggest graph whether frequent and gradually reduce the graph which used to judge until can not mining maximal frequent subgraph, at the same time this algorithm is proposed a theorem and two principles these are improved the mining efficiency.

Editing graphs to satisfy degree constraints: A parameterized approach

We study a wide class of graph editing problems that ask whether a given graph can be modified to satisfy certain degree constraints, using a limited number of vertex deletions, edge deletions, or edge additions. The problems generalize several well-studied problems such as the General Factor Problem and the Regular Subgraph Problem. We classify the parameterized complexity of the considered problems taking upper bounds on the number of editing steps and the maximum degree of the resulting graph as parameters.

A Constant Factor Approximation for Minimum λ-Edge-Connected k-Subgraph with Metric Costs

In the -subgraph problem, we are given an undirected graph with edge costs and two positive integers and , and the goal is to find a minimum cost simple -edge-connected subgraph of with at least nodes. This generalizes several classical problems, such as the minimum cost -spanning tree problem, or -MST (which is a -subgraph), and the minimum cost -edge-connected spanning subgraph (which is a -subgraph). The only previously known results on this problem [L. C. Lau, J. S. Naor, M. R. Salavatipour and M. Singh, SIAM J. Comput., 39 (2009), pp. 1062–1087], [C. Chekuri and N. Korula, in Proceedings of the IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Bangalore, India, LIPIcs 2, Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2008, pp. 119–130] show that the -subgraph problem has an -approximation (even for 2-node-connectivity) and that the -subgraph problem in general is almost as hard as the densest -subgraph problem. In this paper we show that if the edge costs are metric (i.e., satisfy the triangle inequality), like in the -MST problem, then there is an -approximation algorithm for the -subgraph problem. This essentially generalizes the -MST constant factor approximability to higher connectivity.

Degree Bounded Network Design with Metric Costs

Given a complete undirected graph, a cost function on edges, and a degree bound $B$, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree $B$ satisfying given connectivity requirements. Even for a simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a $(2+\frac{1}{k})$-approximation algorithm for the minimum bounded degree $k$-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when $r_{\max}$ is even, and a 5.5-approximation algorithm when $r_{\max}$ is odd, where $r_{\max}$ is the maximum connectivity requirement. In global vertex-connectivity, there is a $(2+\frac{k-1}{n}+\frac{1}{k})$-approximation algorithm for the minimum bounded degree $k$-vertex-connected subgraph problem when $n\geq2k$, where $n$ is the number of vertices. For spanning tree, there is a $(1+\frac{1}{B-1})$-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides' algorithm for the metric traveling salesman problem. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to “short-cut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant

We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant. In other words, we show that if $\rho$ is the expected fraction of constraints satisfied by a random ordering, then obtaining a $\rho'$ approximation for any $\rho'>\rho$ is UG-hard. For the simplest OCSP, the Maximum Acyclic Subgraph (MAS) problem, this implies that obtaining a $\rho$-approximation for any constant $\rho>1/2$ is UG-hard. Specifically, for every constant $\varepsilon>0$ the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction $(1-\varepsilon)$ of its edges, it is UG-hard to find one with more than $(1/2+\varepsilon)$ of its edges. Note that it is trivial to find an acyclic subgraph with $1/2$ the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of G. The MAS problem has been well studied, and beating the random ordering for MAS has been a basic open problem. An OCSP of arity k is specified by a subset $\Pi\subseteq S_k$ of permutations on $\{1,2,\dots,k\}$. An instance of such an OCSP is a set V and a collection of constraints, each of which is an ordered k-tuple of V. The objective is to find a global linear ordering of V while maximizing the number of constraints ordered as in $\Pi$. A random ordering of V is expected to satisfy a $\rho=\frac{|\Pi|}{k!}$ fraction. We show that, for any fixed k, it is hard to obtain a $\rho'$-approximation for $\Pi$-OCSP for any $\rho'>\rho$. The result is in fact stronger: we show that for every $\Lambda\subseteq\Pi\subseteq S_k$, and an arbitrarily small $\varepsilon$, it is hard to distinguish instances where a $(1-\varepsilon)$ fraction of the constraints can be ordered according to $\Lambda$ from instances where at most a $(\rho+\varepsilon)$ fraction can be ordered as in $\Pi$. A special case of our result is that the Betweenness problem is hard to approximate beyond a factor $1/3$. The results naturally generalize to OCSPs which assign a payoff to the different permutations. Finally, our results imply (unconditionally) that a simple semidefinite relaxation for MAS does not suffice to obtain a better approximation.

Sparse Balanced Partitions and the Complexity of Subgraph Problems

We consider the problem of partitioning the vertices of an -vertex graph with maximum degree into classes of size at most in a way that minimizes the number of pairs for which there is an edge between and . We show that there is always such a partition with adjacent pairs, and this bound is tight. This problem is related to questions about the depth of certain graph embeddings, which have been used in the study of the complexity of subgraph and constraint satisfaction problems. (A corrected version of this paper has been appended to the originally posted pdf.)

On-line computation and maximum-weighted hereditary subgraph problems

In this paper1 we study the on-line version of maximum-weighted hereditary subgraph problems. In our on-line model, the final instance (a graph with n vertices) is revealed in t clusters, 2 ? t ? n . We first focus on an on-line version of the maximumweighted hereditary subgraph problem. Then, we deal with the particular case of the independent set problem. We are interested in two types of results: the competitive ratio guaranteed by the on-line algorithm and hardness results that account for the difficulty of the problems and for the quality of algorithms developed to solve them.

Improved approximation bounds for the minimum rainbow subgraph problem

In this paper we consider the Minimum Rainbow Subgraph problem (MRS): Given a graph G with n vertices whose edges are coloured with p colours, find a subgraph F⊆G of minimum order and with p edges such that F contains each colour exactly once.We present a polynomial time (12+(12+ϵ)Δ)-approximation algorithm for the MRS problem for an arbitrary small positive ϵ. This improves the previously best known approximation ratio of 56Δ. We also prove the MRS problem to be NP-hard and APX-hard for graphs with maximum degree 2. Finally we present an algorithm to find an optimal solution in running time O(2(p+2plog2Δ)nO(1)).

Approximating survivable networks with β-metric costs

The Survivable Network Design ( SND) problem seeks a minimum-cost subgraph that satisfies prescribed node-connectivity requirements. We consider SND on both directed and undirected complete graphs with β- metric costs when c ( x z ) ⩽ β [ c ( x y ) + c ( y z ) ] for all x , y , z ∈ V , which varies from uniform costs ( β = 1 / 2 ) to metric costs ( β = 1 ). For the k- Connected Subgraph ( k- CS) problem our ratios are: 1 + 2 β k ( 1 − β ) − 1 2 k − 1 for undirected graphs, and 1 + 4 β 3 k ( 1 − 3 β 2 ) − 1 2 k − 1 for directed graphs and 1 2 ⩽ β < 1 3 . For undirected graphs this improves the ratios β 1 − β of Böckenhauer et al. (2008) [3] and 2 + β k n of Kortsarz and Nutov (2003) [11] for all k ⩾ 4 and 1 2 + 3 k − 2 2 ( 4 k 2 − 7 k + 2 ) ⩽ β ⩽ k 2 ( k + 1 ) 2 − 2 . We also show that SND admits the ratios 2 β 1 − β for undirected graphs, and 4 β 3 1 − 3 β 2 for directed graphs with 1 / 2 ⩽ β < 1 / 3 . For two important particular cases of SND, so-called Subset k- CS and Rooted SND, our ratios are 2 β 3 1 − 3 β 2 for directed graphs and β 1 − β for subset k- CS on undirected graphs.