In this work we provide a decomposition theorem for the class of quaternary and non-binary signed-graphic matroids. This generalizes previous results for binary signed-graphic matroids and graphic matroids, and it provides the theoretical basis for a recognition algorithm.

The development of an efficient and practical method to recognize matrices originating from generalized graphs, in the same way that network matrices are defined over directed graphs, remains an open question. In this paper we present such a recognition procedure for binet matrices which are known to characterize an important class of integral polyhedra. A mathematical programming formulation for the recognition problem is provided, where the special structure of these matrices is expressed via a set of constraints. Finally, given that binet matrices represent certain classes of matroids, we present a set of areas ranging from combinatorial optimization to matroid theory that can benefit from our approach.

In this paper, we present a new method for detecting overlapping communities in networks with a predefined number of clusters called LPAM (Link Partitioning Around Medoids). The overlapping communities in the graph are obtained by detecting the disjoint communities in the associated line graph employing link partitioning and partitioning around medoids which are done through the use of a distance function defined on the set of nodes. We consider both the commute distance and amplified commute distance as distance functions. The performance of the LPAM method is evaluated with computational experiments on real life instances, as well as synthetic network benchmarks. For small and medium-size networks, the exact solution was found, while for large networks we found solutions with a heuristic version of the LPAM method.

In this article, our ultimate goal is to transform a graph’s adjacency matrix into a distance matrix. Because cluster density is not observable prior to the actual clustering, our goal is to find a distance whose pairwise minimization will lead to densely connected clusters. Our thesis is centered on the widely accepted notion that strong clusters are sets of vertices with high induced subgraph density. We posit that vertices sharing more connections are closer to each other than vertices sharing fewer connections. This definition of distance differs from the usual shortest-path distance. At the cluster level, our thesis translates into low mean intra-cluster distances, which reflect high densities. We compare three distance measures from the literature. Our benchmark is the accuracy of each measure’s reflection of intra-cluster density, when aggregated (averaged) at the cluster level. We conduct our tests on synthetic graphs, where clusters and intra-cluster density are known in advance. In this article, we restrict our attention to unweighted graphs with no self-loops or multiple edges. We examine the relationship between mean intra-cluster distances and intra-cluster densities. Our numerical experiments show that Jaccard and Otsuka-Ochiai offer very accurate measures of density, when averaged over vertex pairs within clusters.

In this study, we compare distance measures with respect to their ability to capture vertex community structure and the scalability of their computation. Our goal is to find a distance measure which can be used in an aggregate pairwise minimization clustering scheme. The minimization should lead to subsets of vertices with high induced subgraph density. Our definition of distance is rooted in the notion that vertices sharing more connections are closer to each other than vertices which share fewer connections. This definition differs from that of the geodesic distance typically used in graphs. It is based on neighbourhood overlap, not shortest path. We compare four distance measures from the literature and evaluate their accuracy in reflecting intra-cluster density, when aggregated (averaged) at the cluster level. Our tests are conducted on synthetic graphs, where clusters and intra-cluster densities are known in advance. We find that amplified commute, Otsuka-Ochiai and Jaccard distances display a consistent inverse relation to intra-cluster density. We also conclude that the computation of amplified commute distance does not scale as well to large graphs as that of the other two distances.

With a view on graph clustering, we present a definition of vertex-to-vertex distance which is based on shared connectivity. We argue that vertices sharing more connections are closer to each other than vertices sharing fewer connections. Our thesis is centered on the widely accepted notion that strong clusters are formed by high levels of induced subgraph density, where subgraphs represent clusters. We argue these clusters are formed by grouping vertices deemed to be similar in their connectivity. At the cluster level (induced subgraph level), our thesis translates into low mean intra-cluster distances. Our definition differs from the usual shortest-path geodesic distance. In this article, we compare three distance measures from the literature. Our benchmark is the accuracy of each measure's reflection of intra-cluster density, when aggregated (averaged) at the cluster level. We conduct our tests on synthetic graphs generated using the planted partition model, where clusters and intra-cluster density are known in advance. We examine correlations between mean intra-cluster distances and intra-cluster densities. Our numerical experiments show that Jaccard and Otsuka-Ochiai offer very accurate measures of density, when averaged over vertex pairs within clusters.

Graph clustering, also often referred to as network community detection, is the process of assigning common labels to vertices that are densely connected to each other but sparsely connected to the rest of the graph. There are many different approaches to clustering in the literature. However, in this article, we formulate the clustering problem as a combinatorial optimization problem. Our main contribution is a novel problem formulation that maximizes intra-cluster density, a statistically meaningful quantity. It requires the number of clusters, a softbound on cluster size and a penalty coefficient as parameter inputs. More importantly, it is designed to prevent common degeneracies, like the so-called “mega-clusters”. We end with some suggestions on numerical solution techniques and note that an ensemble-like optimization routine seems promising.

Necessary and sufficient conditions for a cographic matroid having no two special matroids as minors to be signed-graphic are provided. The characterization utilizes the notion of Fournier triples of cocircuits and the recent excluded-minor characterization for regular signed-graphic matroids.

Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results from the theory of matroids by establishing connections to networks by decomposition results of matroids. First, we present the implications of the decomposition of regular matroids to networks and related classes of matrices, and secondly we show that strongly unimodular matrices are closed under k-sums for k = 1, 2 implying a decomposition into highly connected network-representing blocks, which are also shown to have a special structure.

A matroid M has all-graphic cocircuits if M∖Y is a graphic matroid for every cocircuit Y∈C∗(M). In Papalamprou and Pitsoulis (2013) it has been shown that signed-graphic matroids that are representable in GF(2) can be decomposed into graphic matroids and matroids with all-graphic cocircuits. In this paper we study this class of signed-graphic matroids with all-graphic cocircuits and provide the set of regular excluded minors as well as a complete characterization when the matroid is cographic, which in turn results in a simple recognition algorithm.

Given a dataset an outlier can be defined as an observation that does not follow the statistical properties of the majority of the data. Computation of the location estimate is of fundamental importance in data analysis, and it is well known in statistics that classical methods, such as taking the sample average, can be greatly affected by the presence of outliers in the data. Using the median instead of the mean can partially resolve this issue but not completely. For the univariate case, a robust version of the median is the Least Trimmed Absolute Deviation (LTAD) robust estimator introduced in Tableman (Stat Probab Lett 19(5):387–398, 1994), which has desirable asymptotic properties such as robustness, consistently, high breakdown and normality. There are different generalizations of the LTAD for multivariate data, depending on the choice of norm. Chatzinakos et al. (J Comb Optim, 2015) we present such a generalization using the Euclidean norm and propose a solution technique for the resulting combinatorial optimization problem, based on a necessary condition, that results in a highly convergent local search algorithm. In this subsequent work, we use the $$L^1$$ norm to generalize the LTAD to higher dimensions, and show that the resulting mixed integer programming problem has an integral relaxation, after applying an appropriate data transformation. Moreover, we utilize the structure of the problem to show that the resulting LP's can be solved efficiently using a subgradient optimization approach. The robust statistical properties of the proposed estimator are verified by extensive computational results.

Computation of typical statistical sample estimates such as the median or least squares fit usually require the solution of an unconstrained optimization problem with a convex objective function, that can be solved efficiently by various methods. The presence of outliers in the data dictates the computation of a robust estimate, which can be defined as the optimum statistical estimate for a subset that contains at least half of the observations. The resulting problem is now a combinatorial optimization problem which is often computationally intractable. Classical statistical methods for multivariate location \(\varvec{\mu }\) and scatter matrix \(\varvec{\varSigma }\) estimation are based on the sample mean vector and covariance matrix, which are very sensitive in the presence of outlier observations. We propose a new method for robust location and scatter estimation which is composed of two stages. In the first stage an unbiased multivariate \(L_{1}\)-median center for all the observations is attained by a novel procedure called the least trimmed Euclidean deviations estimator. This robust median defines a coverage set of observations which is used in the second stage to iteratively compute the set of outliers which violate the correlational structure of the data set. Extensive computational experiments indicate that the proposed method outperforms existing methods in accuracy, robustness and computational time.

The shortest superstring problem (SSP) is an \(NP\)-hard combinatorial optimization problem which has attracted the interest of many researchers, due to its applications in computational molecular biology problems such as DNA sequencing, and in computer science problems such as string compression. In this paper a new heuristic algorithm for solving large scale instances of the SSP is presented, which outperforms the natural greedy algorithm in the majority of the tested instances. The proposed method is able to provide multiple near-optimum solutions and admits a natural parallel implementation. Extended computational experiments on a set of SSP instances with known optimum solutions indicate that the new method finds the optimum solution in most of the cases, and its average error relative to the optimum is close to zero.