Bipartite Turán problems via graph gluing

Even, Itai, and Shamir (1976) proved simple two‐commodity integral flow is NP‐complete both in the directed and undirected cases. In particular, the directed case was shown to be NP‐complete even if one demand is unitary, which was improved by Fortune, Hopcroft and Wyllie (1980) who proved the problem is still NP‐complete if both demands are unitary. The undirected case, on the other hand, was proved by Robertson and Seymour (1995) to be polynomial‐time solvable if both demands are constant. Nevertheless, the complexity of the undirected case with exactly one constant demand has remained unknown. We close this 40‐year complexity gap, by showing the undirected case is NP‐complete even if exactly one demand is unitary. As a by product, we obtain the NP‐completeness of determining whether a graph contains 1 + d pairwise vertex‐disjoint paths, such that one path is between a given pair of vertices and d paths are between a second given pair of vertices. Additionally, we investigate the complexity of another related network design problem called strict terminal connection.

Simple Undirected Two-Commodity Integral Flow with a Unitary Demand

A 2-dimensional framework (G, p) is a graph G = (V, E) together with a map \(p: V \rightarrow \mathbb{R}^{2}\). We consider the framework to be a straight line realization of G in \(\mathbb{R}^{2}\). Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u, v} is globally linked in G if the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G.In this paper we extend our previous results on globally linked pairs and complete the characterization of globally linked pairs in minimally rigid graphs. We also show that the Henneberg 1-extension operation on a non-redundant edge preserves the property of being not globally linked, which can be used to identify globally linked pairs in broader families of graphs. We prove that if (G, p) is generic then the set of globally linked pairs does not change if we perturb the coordinates slightly. Finally, we investigate when we can choose a non-redundant edge e of G and then continuously deform a generic realization of G − e to obtain equivalent generic realizations of G in which the distances between a given pair of vertices are different.KeywordsGlobal rigidityRigid frameworkBar-and-joint frameworkGlobally linked pairMinimally rigid graphSubject Classifications52C2505C10

Short length Menger's theorem and reliable optical routing

Stress is a centrality measure determined by the shortest paths passing through the given vertex. Noting that adjacency matrix playing an important role in finding the distance and the number of shortest paths between given pair of vertices, an interesting expression and also an algorithm are presented to find stress using adjacency matrix. The results and algorithm are suitably adopted to obtain betweenness centrality measure. Further results are extended to the cases of Cartesian product of graphs, corona graph and their special cases.

In this paper, we study the problems of enumerating cuts of a graph by non-decreasing weights. There are four problems, depending on whether the graph is directed or undirected, and on whether we consider all cuts of the graph or only s-t cuts for a given pair of vertices s, t. Efficient algorithms for these problems with \({\tilde{\mbox{O}}}(n^2m)\) delay between two successive outputs have been known since 1992, due to Vazirani and Yannakakis. In this paper, improved algorithms are presented. The delays of the presented algorithms are O(nm log(n 2/m)).KeywordsUndirected GraphBasic PartitionSink VertexSink SideBasic SubroutineThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Community detection is the problem of identifying community structure in graphs. Often the graph is modeled as a sample from the Stochastic Block Model, in which each vertex belongs to a community. The probability that two vertices are connected by an edge depends on the communities of those vertices. In this paper, we consider a model of {\em censored} community detection with two communities, where most of the data is missing as the status of only a small fraction of the potential edges is revealed. In this model, vertices in the same community are connected with probability $p$ while vertices in opposite communities are connected with probability $q$. The connectivity status of a given pair of vertices $\{u,v\}$ is revealed with probability $α$, independently across all pairs, where $α= \frac{t \log(n)}{n}$. We establish the information-theoretic threshold $t_c(p,q)$, such that no algorithm succeeds in recovering the communities exactly when $t < t_c(p,q)$. We show that when $t > t_c(p,q)$, a simple spectral algorithm based on a weighted, signed adjacency matrix succeeds in recovering the communities exactly. While spectral algorithms are shown to have near-optimal performance in the symmetric case, we show that they may fail in the asymmetric case where the connection probabilities inside the two communities are allowed to be different. In particular, we show the existence of a parameter regime where a simple two-phase algorithm succeeds but any algorithm based on the top two eigenvectors of the weighted, signed adjacency matrix fails.

A machine learning approach for predicting human shortest path task performance

A characterization of some distance-regular graphs by strongly closed subgraphs

We introduce the problem of clustering the set of vertices in a given 3D mesh. The problem is motivated by the need for value engineering in architectural projects. We first derive a max-norm based metric to estimate the geometric disparity between a given pair of vertices, and characterize the problem in terms of this measure. We show that this distance can be computed by using Sequential Quadratic Programming (SQP). Next we introduce two different algorithms for clustering the set of vertices on a given mesh, respectively based on two disparity measurements: max-norm and L2-norm based metric. An equivalence is established between mesh vertices and physical joints in an architectural mesh. By replacing individual joints by their equivalent cluster representative, the number of unique joints in the facade mesh, and therefore the fabrication cost, is dramatically reduced. Finally, we present an algorithm for remeshing a given surface in order to further reduce the number of joint clusters. The framework is tested for a set of real-world architectural surfaces to illustrate the effectiveness and utility of our approach. Overall, this approach tackles the important problem reducing fabrication cost of joints without modifying the underlying connectivity that was specified by the architect.

In this paper, we study the problems of enumerating cuts of a graph by non-decreasing weights. There are four problems, depending on whether the graph is directed or undirected, and on whether we consider all cuts of the graph or only s-t cuts for a given pair of vertices s,t. Efficient algorithms for these problems with $\tilde{O}(n^{2}m)$ delay between two successive outputs have been known since 1992, due to Vazirani and Yannakakis. In this paper, improved algorithms are presented. The delays of the presented algorithms are O(nmlogź(n2/m)). Vazirani and Yannakakis's algorithms have been used as basic subroutines in the solutions of many problems. Therefore, our improvement immediately reduces the running time of these solutions. For example, for the minimum k-cut problem, the upper bound is immediately reduced by a factor of $\tilde{O}(n)$ for k=3,4,5,6.

Consider a setting where possibly sensitive information sent over a path in a network is visible to every neighbor of (some node on) the path, thus including the nodes on the path itself. The exposure of a path P can be measured as the number of nodes adjacent to it, denoted by N[P]. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected n-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of \(O(2^{\log^{1-\epsilon}n})\) for any ε > 0 (under an appropriate complexity assumption), but is approximable with ratio \(\sqrt{\Delta }+3\), where Δ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial time algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs. Turning to the more general problem of finding a minimum exposure Steiner tree connecting a given set of k terminals, the picture becomes more involved. In undirected unweighted graphs with unbounded degree, we present an approximation algorithm with ratio \(\min\{\Delta , n/k, \sqrt{2n},O(\log k \cdot (k+\sqrt{\Delta }))\}\). On unweighted undirected bounded-degree graphs, the problem is still polynomial when the number of terminals is fixed, but if the number of terminals is arbitrary, then the problem becomes NP-hard again.KeywordsOptimal PathConnected SubgraphSteiner Tree ProblemConnectivity ProblemBarrier CoverageThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Consider a setting where possibly sensitive information sent over a path in a network is visible to every neighbor of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path P can be measured as the number of nodes adjacent to it, denoted by N[P]. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected n-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $$O(2^{\log ^{1-\epsilon }n})$$ for any $$\epsilon >0$$ (under an appropriate complexity assumption), but is approximable with ratio $$\sqrt{\Delta }+3$$ , where $$\Delta $$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs. Turning to the more general problem of finding a minimum exposure Steiner tree connecting a given set of k terminals, the picture becomes more involved. In undirected unweighted graphs with unbounded degree, we present an approximation algorithm with ratio $$\min \{\Delta , n/k, \sqrt{2n},O(\log k \cdot (k+\sqrt{\Delta }))\}$$ . On unweighted undirected bounded-degree graphs, the problem is still polynomial when the number of terminals is fixed, but if the number of terminals is arbitrary, then the problem becomes NP-hard again.

Computing strictly-second shortest paths

Given a graph \(G=(V,E)\) with two distinguished vertices \(s,t\in V\) and an integer L, an L-bounded flow is a flow between s and t that can be decomposed into paths of length at most L. In the maximum L-bounded flow problem the task is to find a maximum L-bounded flow between a given pair of vertices in the input graph.