Problem For Undirected Graphs Research Articles

Vertex-connectivity for node failure identification in Boolean Network Tomography

We study the node failure identification problem in undirected graphs by means of Boolean Network Tomography. We argue that vertex-connectivity plays a central role. We prove bounds on the maximum number of simultaneous node failures that can be identified in arbitrary networks. We argue that (augmented) grids are a class of networks with large failure identifiability, and provide very tight results in this context.

Modelling conflicting individual preference: Target sequences and graph realization

<p style='text-indent:20px;'>This paper will consider a group of individuals who each have a target number of contacts they would like to have with other group members. We are interested in how close this can be to being realized, considering the group's long-term outcome under reasonable dynamics on the number of contacts. We formulate this as a graph realization problem for undirected graphs, with the individuals as vertices and the number of desired contacts as the vertex degree. It is well known that not all degree sequences can be realized as undirected graphs, and the Havel-Hakimi algorithm characterizes those that can. When we ask how close the degree sequences can be to realization, we ask for graphs that minimize the total deviation between what is desired and possible. The sets of all such graphs and all such associated sequences are termed the minimal sets. Broom and Cannings have previously considered this problem in many papers, and it is hard to tackle for general target sequences. This paper revisited the minimal set in general, investigating two particular classes of sequence in particular. We consider the n-element arithmetic sequence (n-1, n-2, … 1, 0) for general n, including obtaining a formula that generates the size of the minimal set for a given arithmetic sequence, and the all or nothing sequences, where targets are either 0 or n-1, where a recurrence relation for such a formula is found. Further, we consider the question of the size of the minimal set of sequences in general. We consider a strategic version of the model where the individuals are involved in a multiplayer game, each trying to achieve their target, and show that optimal play can lead to the minimal set being left, thus answering an open question from earlier work.</p>

Parameterized Complexity of Directed Spanner Problems

We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most m-k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most m-k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size mathcal {O}(k^4t^5) and can be solved in randomized (4t)^kcdot n^{mathcal {O}(1)} time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in k^{2k}cdot n^{mathcal {O}(1)} time on directed acyclic graphs, (iii) Directed Additive Spanner is {{,mathrm{mathsf{W}},}}[1]-hard when parameterized by k for every fixed tge 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is {{,mathrm{mathsf{FPT}},}} when parameterized by t and k.

Multi-start iterated local search, exact and matheuristic approaches for minimum capacitated dominating set problem

This paper presents a multi-start iterated local search (MS-ILS), an exact approach based on an improved integer linear programming (ILP) model, and a matheuristic combining the ILP approach with MS-ILS through solution merging for the minimum capacitated dominating set (CAPMDS) problem for undirected graphs. This problem is an extension of the well-known minimum dominating set problem, where there is a cap on the number of nodes that each node can dominate, and the goal is to find a dominating set of minimum cardinality where the number of dominated nodes for each dominating node is within this cap. This NP-hard problem has several real world applications in resource constrained environments such as clustering of wireless sensor networks along with selection of cluster heads, document summarization in information retrieval. Computational results on the standard benchmark instances of this problem show the superiority of our approaches over state-of-the-art approaches available in the literature in their respective categories.

Barrakuda: A Hybrid Evolutionary Algorithm for Minimum Capacitated Dominating Set Problem

The minimum capacitated dominating set problem is an NP-hard variant of the well-known minimum dominating set problem in undirected graphs. This problem finds applications in the context of clustering and routing in wireless networks. Two algorithms are presented in this work. The first one is an extended version of construct, merge, solve and adapt, while the main contribution is a hybrid between a biased random key genetic algorithm and an exact approach which we labeled Barrakuda. Both algorithms are evaluated on a large set of benchmark instances from the literature. In addition, they are tested on a new, more challenging benchmark set of larger problem instances. In the context of the problem instances from the literature, the performance of our algorithms is very similar. Moreover, both algorithms clearly outperform the best approach from the literature. In contrast, Barrakuda is clearly the best-performing algorithm for the new, more challenging problem instances.

Finding a path with two labels forbidden in group-labeled graphs

The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where “non-zero” means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an s – t path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero s – t path in a Z 3 -labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s – t paths in a group-labeled graph or not, and finding s – t paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s – t paths, which is the main technical contribution of this paper.

Fully dynamic depth-first search in directed graphs

Depth-first search (DFS) is a fundamental and important algorithm in graph analysis. It is the basis of many graph algorithms such as computing strongly connected components, testing planarity, and detecting biconnected components. The result of a DFS is normally shown as a DFS-Tree. Given the frequent updates in many real-world graphs (e.g., social networks and communication networks), we study the problem of DFS-Tree maintenance in dynamic directed graphs. In the literature, most works focus on the DFS-Tree maintenance problem in undirected graphs and directed acyclic graphs. However, their methods cannot easily be applied in the case of general directed graphs. Motivated by this, we propose a framework and corresponding algorithms for both edge insertion and deletion in general directed graphs. We further give several optimizations to speed up the algorithms. We conduct extensive experiments on 12 real-world datasets to show the efficiency of our proposed algorithms.

On the Shoshan-Zwick Algorithm for the All-Pairs Shortest Path Problem

The Shoshan-Zwick algorithm solves the all pairs shortest paths problem in undirected graphs with integer edge costs in the range $\{1, 2, \dots, M\}$. It runs in $\tilde{O}(M\cdot n^{\omega})$ time, where $n$ is the number of vertices, $M$ is the largest integer edge cost, and $\omega < 2.3727$ is the exponent of matrix multiplication. It is the fastest known algorithm for this problem. This paper points out the erroneous behavior of the Shoshan-Zwick algorithm and revises the algorithm to resolve the issues that cause this behavior. Moreover, it discusses implementation aspects of the Shoshan-Zwick algorithm using currently-existing sub-cubic matrix multiplication algorithms.

An optimal adaptive stabilizing algorithm for two edge-disjoint paths problem

The problem of two edge-disjoint paths is to identify two paths $$Q_1$$Q1 and $$Q_2$$Q2 from source $$s \in V$$s?V to target $$t \in V$$t?V without any common arc in a directed connected graph $$G=(V, E)$$G=(V,E). In this paper, we present an adaptive stabilizing algorithm for finding a pair of edge-disjoint paths from s to t in G in O(D) rounds with state-space complexity of $$O(log\; n)$$O(logn) bits per process, where n is the number of nodes and D is the diameter of the graph. The proposed algorithm is optimal with respect to its time complexity, and the total length of the shortest paths. In addition, it can also be used to solve the problem for undirected graphs. Since the proposed algorithm is stabilizing, it does not require initialization and is capable of withstanding transient faults. We view a fault that perturbs the state of the system but not its program as a transient fault. In addition, the proposed algorithm is adaptive since it is capable of dealing with topology changes in the form of addition/removal of arcs and/or nodes as well as changes in the directions of arcs provided that two edge-disjoint paths between s and t exist after the topology change.

Strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs

We consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded-rank hypergraphs with multiple edge cost functions. For a fixed number of edge cost functions, we show that the total number of supported non-dominated (SND) cuts is bounded by a polynomial in the numbers of nodes and edges, i.e., is strongly polynomial. This bound also applies to the combinatorial facet complexity of the problem, i.e., the maximum number of facets (linear pieces) of the parametric curve for the parametrized (linear combination) objective, over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive. We sharpen this bound in the case of two objectives (the bicriteria problem), for which we also derive a strongly polynomial upper bound on the total number of non-dominated (Pareto optimal) cuts. In particular, the bicriteria global minimum cut problem in an n-node graph admits $$O(n^3 \log n)$$O(n3logn) SND cuts and $$O(n^5 \log n)$$O(n5logn) Pareto optimal cuts. These results significantly improve on earlier graph cut results by Mulmuley (SIAM J Comput 28(4):1460---1509, 1999) and Armon and Zwick (Algorithmica 46(1):15---26, 2006). They also imply that the parametric curve and all SND cuts, and, for the bicriteria problems, all Pareto optimal cuts, can be computed in strongly polynomial time when the number of objectives is fixed.

The minimum vulnerability problem on specific graph classes

Suppose that each edge e of an undirected graph G is associated with three nonnegative integers $$\mathsf{cost}(e)$$cost(e), $$\mathsf{vul}(e)$$vul(e) and $$\mathsf{cap}(e)$$cap(e), called the cost, vulnerability and capacity of e, respectively. Then, we consider the problem of finding $$k$$k paths in G between two prescribed vertices with the minimum total cost; each edge e can be shared without any cost by at most $$\mathsf{vul}(e)$$vul(e) paths, and can be shared by more than $$\mathsf{vul}(e)$$vul(e) paths if we pay $$\mathsf{cost}(e)$$cost(e), but cannot be shared by more than $$\mathsf{cap}(e)$$cap(e) paths even if we pay the cost for e. This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number $$k$$k of required paths.

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Given a graph G and an integer k , the Feedback Vertex Set (FVS) problem asks if there is a vertex set T of size at most k that hits all cycles in the graph. The first fixed-parameter algorithm for FVS in undirected graphs appeared in a monograph of Mehlhorn in 1984. The fixed-parameter tractability (FPT) status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC ’08, JACM ’08) showed that it is fixed-parameter tractable by giving a 4 k k ! · n O (1) time algorithm. There are two subset versions of this problems: We are given an additional subset S of vertices (resp., edges), and we want to hit all cycles passing through a vertex of S (resp., an edge of S ); the two variants are known to be equivalent in the parameterized sense. Recently, the Subset FVS problem in undirected graphs was shown to be FPT by Cygan et al. (ICALP’11, SIDMA’13) and independently by Kakimura et al. (SODA ’12). We generalize the result of Chen et al. (STOC ’08, JACM ’08) by showing that a Subset FVS in directed graphs can be solved in time 2 O ( k 3 ) ċ n O (1) (i.e., FPT parameterized by size k of the solution). By our result, we complete the picture for FVS problems and their subset versions in undirected and directed graphs. The technique of random sampling of important separators was used by Marx and Razgon (STOC ’11, SICOMP ’14) to show that Undirected Multicut is FPT, and it was generalized by Chitnis et al. (SODA ’12, SICOMP ’13) to directed graphs to show that Directed Multiway Cut is FPT. In addition to proving the FPT of a Directed Subset FVS, we reformulate the random sampling of important separators technique in an abstract way that can be used with a general family of transversal problems. We believe this general approach will be useful for showing the FPT of other problems in directed graphs. Moreover, we modify the probability distribution used in the technique to achieve better running time; in particular, this gives an improvement from 2 2 O ( k ) to 2 O ( k 2 ) in the parameter dependence of the Directed Multiway Cut algorithm of Chitnis et al. (SODA ’12, SICOMP ’13).

On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for some class of finite-dimensional 2-step nilpotent Lie algebras over a field and for some class of finite p-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J. D. Hamkins, and R. Miller (2012). A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied.

Approximating Rooted Steiner Networks

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques, we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω( k ϵ ) hardness bound for the rooted k -connectivity problem in undirected graphs. As a consequence, we obtain an Ω( k ϵ ) hardness bound for the undirected subset k -connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k -connectivity problem.

A new algorithm for the minimum spanning tree verification problem

This paper proposes a new algorithm for the minimum spanning tree verification (MSTV) problem in undirected graphs. The MSTV problem is distinct from the minimum spanning tree construction problem. The above problems have been studied extensively, and there exist several papers in the literature devoted to them. Our algorithm for the MSTV problem combines the insights of Bor?vka's algorithm for constructing a minimum spanning tree with a bucketing scheme. The principal idea underlying this combination is the efficient identification of edges that cannot be part of any minimum spanning tree. Although the proposed algorithm imposes no restrictions on the input graph, it was designed to exploit the case in which the number of distinct edge weights is small. On a graph with $$n$$n vertices, $$m$$m edges, and $$K$$K distinct edge weights, our algorithm runs in $$O(m+n\cdot K)$$O(m+n·K) time. It follows that our algorithm runs in linear time if $$K$$K is a fixed constant. Although there exist several linear time MSTV algorithms in the literature, our algorithm improves on the state of the art in two ways, viz., it is conceptually simpler, and it is easy to implement. We contrast the performance of our algorithm vis-a-vis Hagerup's linear time MSTV algorithm, which is one of the more practical linear-time MSTV algorithms. Our experiments indicate that the proposed algorithm is superior to Hagerup's algorithm when $$K \le 24$$K≤24. One surprising observation is that our algorithm is substantially faster than Hagerup's algorithm on "No" instances, i.e., on instances in which the input spanning tree is not the minimum spanning tree.

Algorithms for finding maximum transitive subtournaments

The problem of finding a maximum clique is a fundamental problem for undirected graphs, and it is natural to ask whether there are analogous computational problems for directed graphs. Such a problem is that of finding a maximum transitive subtournament in a directed graph. A tournament is an orientation of a complete graph; it is transitive if the occurrence of the arcs $$xy$$xy and $$yz$$yz implies the occurrence of $$xz$$xz. Searching for a maximum transitive subtournament in a directed graph $$D$$D is equivalent to searching for a maximum induced acyclic subgraph in the complement of $$D$$D, which in turn is computationally equivalent to searching for a minimum feedback vertex set in the complement of $$D$$D. This paper discusses two backtrack algorithms and a Russian doll search algorithm for finding a maximum transitive subtournament, and reports experimental results of their performance.

Obtaining Matrices with the Consecutive Ones Property by Row Deletions

A binary matrix $$M$$M has the Consecutive Ones Property (COP) if there exists a permutation of columns that arranges the ones consecutively in all the rows. We consider the parameterized complexity of $$d$$d-COS-R (Consecutive Ones Submatrix by Row deletions) problem [9]: Given a matrix $$M$$M and a positive integer $$d$$d, decide whether there exists a set of at most $$d$$d rows of $$M$$M whose deletion results in a matrix with the COP. The closely related Interval Deletion problem has recently been shown to be FPT [6]. We show that $$d$$d-COS-R is fixed-parameter tractable and has the current best run-time of $$O^*(10^d)$$O?(10d), which is associated with the Interval Deletion problem. We also introduce a closely related optimization problem called Min-ICPSA: For a finite sized universe $$\mathcal{U}$$U the input consists of a family of $$n$$n pairs of sets $$\mathcal{S} = \{(A_i,B_i) \mid A_i, B_i \subseteq \mathcal{U}, 1 \le i \le n\}$$S={(Ai,Bi)?Ai,Bi⊆U,1≤i≤n}; the aim is to find a minimum number of pairs of sets to discard from $$\mathcal{S}$$S such that for each one of the remaining pairs, say $$(A_k,B_k), |A_k|=|B_k|$$(Ak,Bk),|Ak|=|Bk|, and for any two of the remaining pairs, indexed by $$1 \le j \ne k \le n$$1≤j?k≤n, $$|A_j \cap A_k| = |B_j \cap B_k|$$|Aj?Ak|=|Bj?Bk|. We show that Min-ICPSA is computationally equivalent to the Vertex Cover problem. We also show that it is at least as hard as the Hamilton Path problem in undirected graphs, even when each $$|A_k|=2, 1 \le k \le n$$|Ak|=2,1≤k≤n.

Optimal shortest path set problem in undirected graphs

This paper proposes the optimal shortest path set problem in an undirected graph $$G=(V,E)$$ G = ( V , E ) , in which some vehicles have to go from a source node $$s$$ s to a destination node $$t$$...

Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs

This technical note studies the continuous-time distributed optimization of a sum of convex functions over directed graphs. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and directed scenarios, we show that the consensus-based dynamics that solves the continuous-time distributed optimization problem for undirected graphs fails to converge when transcribed to the directed setting. This study sets the basis for the design of an alternative distributed dynamics which we show is guaranteed to converge, on any strongly connected weight-balanced digraph, to the set of minimizers of a sum of convex differentiable functions with globally Lipschitz gradients. Our technical approach combines notions of invariance and cocoercivity with the positive definiteness properties of graph matrices to establish the results.

Optimal Matching Forests and Valuated Delta-Matroids

The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an $\mathrm{O}(n^{2}m)$-time algorithm for finding a maximum-weight matching forest, where $n$ is the number of vertices and $m$ is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total dual integrality of the linear system. In the present paper, we reveal another nice property of matching forests: the degree sequences of the matching forests in any mixed graph form a delta-matroid, and the weighted matching forests induce a valuated delta-matroid. We remark that the delta-matroid is not necessarily even, and the valuated delta-matroid induced by weighted matching forests slightly generalizes the well-known notion of Dress and Wenzel's valuated delta-matroids. By focusing on the delta-matroid structure and reviewing Giles' algorithm, we design a simpler $\mathrm{O}(n^{2}m)$-time ...