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We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n \operatorname{polylog} n)$ total time. Interspersed with the deletions, it can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental $c$-edge-connectivity to a variant of fully-dynamic $c$-edge-connectivity on a sparse graph. While being Monte-Carlo, our reduction supports a certain final self-check that can be used in Las Vegas algorithms for static problems such as Unique Perfect Matching. For non-sparse graphs with $\Omega(n \operatorname{polylog} n)$ edges, our connectivity and $2$-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an $O(m \log (n^2 / m) + n \operatorname{polylog} n)$-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with $\Omega(n^2)$ edges. Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an $\Omega(\log n/\log\log n)$ worst-case time lower bound in the decremental setting [Alstrup, Thore Husfeldt, FOCS'98] as well as an $\Omega(\log n)$ amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].

Inverse optimization is the problem of determining the values of missing input parameters for an associated forward problem that are closest to given estimates and that will make a given target vector optimal. This study is concerned with the relationship of a particular inverse mixed integer linear optimization problem (MILP) to both the forward problem and the separation problem associated with its feasible region. We show that a decision version of the inverse MILP in which a primal bound is verified is coNP-complete, whereas primal bound verification for the associated forward problem is NP-complete, and that the optimal value verification problems for both the inverse problem and the associated forward problem are complete for the complexity class D^P. We also describe a cutting-plane algorithm for solving inverse MILPs that illustrates the close relationship between the separation problem for the convex hull of solutions to a given MILP and the associated inverse problem. The inverse problem is shown to be equivalent to the separation problem for the radial cone defined by all inequalities that are both valid for the convex hull of solutions to the forward problem and binding at the target vector. Thus, the inverse, forward, and separation problems can be said to be equivalent.

Factor graph of an instance of a constraint satisfaction problem with n variables and m constraints is the bipartite graph between [m] and [n] describing which variable appears in which constraints. Thus, an instance of a CSP is completely defined by its factor graph and the list of predicates. We show inapproximability of Max-3-LIN over non-abelian groups (both in the perfect completeness case and in the imperfect completeness case), with the same inapproximability factor as in the general case, even when the factor graph is fixed. Along the way, we also show that these optimal hardness results hold even when we restrict the linear equations in the Max-3-LIN instances to the form x * y * z = g, where x, y, z are the variables and g is a group element. We use representation theory and Fourier analysis over non-abelian groups to analyze the reductions.

Principal Component Analysis (PCA) and other multi-variate models are often used in the analysis of "omics" data. These models contain much information which is currently neither easily accessible nor interpretable. Here we present an algorithmic method which has been developed to integrate this information with existing databases of background knowledge, stored in the form of known sets (for instance genesets or pathways). To make this accessible we have produced a Graphical User Interface (GUI) in Matlab which allows the overlay of known set information onto the loadings plot and thus improves the interpretability of the multi-variate model. For each known set the optimal convex hull, covering a subset of elements from the known set, is found through a search algorithm and displayed. In this paper we discuss two main topics; the details of the search algorithm for the optimal convex hull for this problem and the GUI interface which is freely available for download for academic use.

We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open question of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars. Furthermore, we introduce a vertex-variant of interval-membership-width (which can be arbitrarily larger than its edge-counterpart) and use it to obtain an FPT-time algorithm for a natural vertex-exploration problem that remains hard even when interval-membership-width is bounded.

The inverse geodesic length of a graph $G$ is the sum of the inverse of the distances between all pairs of distinct vertices of $G$. In some domains it is known as the Harary index or the global efficiency of the graph. We show that, if $G$ is planar and has $n$ vertices, then the inverse geodesic length of $G$ can be computed in roughly $O(n^{9/5})$ time. We also show that, if $G$ has $n$ vertices and treewidth at most $k$, then the inverse geodesic length of $G$ can be computed in $O(n \log^{O(k)}n)$ time. In both cases we use techniques developed for computing the sum of the distances, which does not have "inverse" component, together with batched evaluations of rational functions.

We give an algorithm to find a minimum cut in an edge-weighted directed graph with $n$ vertices and $m$ edges in $\tilde O(n\cdot \max(m^{2/3}, n))$ time. This improves on the 30 year old bound of $\tilde O(nm)$ obtained by Hao and Orlin for this problem. Our main technique is to reduce the directed mincut problem to $\tilde O(\min(n/m^{1/3}, \sqrt{n}))$ calls of {\em any} maxflow subroutine. Using state-of-the-art maxflow algorithms, this yields the above running time. Our techniques also yield fast {\em approximation} algorithms for finding minimum cuts in directed graphs. For both edge and vertex weighted graphs, we give $(1+\epsilon)$-approximation algorithms that run in $\tilde O(n^2 / \epsilon^2)$ time.

Let $G$ be a graph with vertex set $V$. {Two disjoint sets $V_1, V_2 \subseteq V$ form a coalition in $G$ if none of them is a dominating set of $G$ but their union $V_1\cup V_2$ is. A vertex partition $\Psi=\{V_1,\ldots, V_k\}$ of $V$ is called a coalition partition of $G$ if every set~$V_i\in \Psi$ is either a dominating set of $G$ with the cardinality $|V_i|=1$, or is not a dominating set but for some $V_j\in \Psi$, $V_i$ and $V_j$ form a coalition.} The maximum cardinality of a coalition partition of $G$ is called the coalition number of $G$, denoted by $\mathcal{C}(G)$. A $\mathcal{C}(G)$-partition is a coalition partition of $G$ with cardinality $\mathcal{C}(G)$. Given a coalition partition $\Psi=\{V_1, V_2,\ldots, V_r\}$ of~$G$, a coalition graph $CG(G, \Psi)$ is associated on $\Psi$ such that there is a one-to-one correspondence between its vertices and the members of $\Psi$. Two vertices of $CG(G, \Psi)$ are adjacent if and only if the corresponding sets form a coalition in $G$. In this paper, we first show that for any graph $G$ with $\delta(G)=1$, $\mathcal{C}(G)\leq 2\Delta(G)+2$, where $\delta(G)$ and $\Delta(G)$ are the minimum degree and the maximum degree of $G$, respectively. Moreover, we characterize all graphs~$G$ with $\delta(G)\leq 1$ and $\mathcal{C}(G)=n$, where $n$ is the number of vertices of $G$. Furthermore, we characterize all trees $T$ with $\mathcal{C}(T)=n$ and all trees $T$ with $\mathcal{C}(T)=n-1$. This solves partially one of the open problem posed in~\cite{coal0}. On the other hand, we theoretically and empirically determine the number of coalition graphs that can be defined by all coalition partitions of a given path $P_k$. Furthermore, we show that there is no universal coalition path, a path whose coalition partitions defines all possible coalition graphs. These solve two open problems posed by Haynes et al.~\cite{coal1}.

Graph Convolution Networks (GCNs) manifest great potential in recommendation. This is attributed to their capability on learning good user and item embeddings by exploiting the collaborative signals from the high-order neighbors. Like other GCN models, the GCN based recommendation models also suffer from the notorious over-smoothing problem - when stacking more layers, node embeddings become more similar and eventually indistinguishable, resulted in performance degradation. The recently proposed LightGCN and LR-GCN alleviate this problem to some extent, however, we argue that they overlook an important factor for the over-smoothing problem in recommendation, that is, high-order neighboring users with no common interests of a user can be also involved in the user's embedding learning in the graph convolution operation. As a result, the multi-layer graph convolution will make users with dissimilar interests have similar embeddings. In this paper, we propose a novel Interest-aware Message-Passing GCN (IMP-GCN) recommendation model, which performs high-order graph convolution inside subgraphs. The subgraph consists of users with similar interests and their interacted items. To form the subgraphs, we design an unsupervised subgraph generation module, which can effectively identify users with common interests by exploiting both user feature and graph structure. To this end, our model can avoid propagating negative information from high-order neighbors into embedding learning. Experimental results on three large-scale benchmark datasets show that our model can gain performance improvement by stacking more layers and outperform the state-of-the-art GCN-based recommendation models significantly.

The Normalized Cut (NCut) objective function, widely used in data clustering and image segmentation, quantifies the cost of graph partitioning in a way that biases clusters or segments that are balanced towards having lower values than unbalanced partitionings. However, this bias is so strong that it avoids any singleton partitions, even when vertices are very weakly connected to the rest of the graph. Motivated by the B\"uhler-Hein family of balanced cut costs, we propose the family of Compassionately Conservative Balanced (CCB) Cut costs, which are indexed by a parameter that can be used to strike a compromise between the desire to avoid too many singleton partitions and the notion that all partitions should be balanced. We show that CCB-Cut minimization can be relaxed into an orthogonally constrained $\ell_{\tau}$-minimization problem that coincides with the problem of computing Piecewise Flat Embeddings (PFE) for one particular index value, and we present an algorithm for solving the relaxed problem by iteratively minimizing a sequence of reweighted Rayleigh quotients (IRRQ). Using images from the BSDS500 database, we show that image segmentation based on CCB-Cut minimization provides better accuracy with respect to ground truth and greater variability in region size than NCut-based image segmentation.