309 research outputs found

    Long Circuits and Large Euler Subgraphs

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    An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is well known that Eulerian graphs can be recognized in polynomial time while the problems of finding a maximum Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this paper, we study the parameterized complexity of the following Euler subgraph problems: - Large Euler Subgraph: For a given graph G and integer parameter k, does G contain an induced Eulerian subgraph with at least k vertices? - Long Circuit: For a given graph G and integer parameter k, does G contain an Eulerian subgraph with at least k edges? Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. We find this a bit surprising because the problem of finding an induced Eulerian subgraph with exactly k vertices is known to be W[1]-hard. The complexity of the problem changes drastically on directed graphs. On directed graphs we obtained the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed k>3 and is solvable in polynomial time for k<=3. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs

    Preventing Unraveling in Social Networks Gets Harder

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    The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp} introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree kk the equilibrium for this model is a maximal subgraph whose minimum degree is β‰₯k\geq k. However the dropping out of individuals with degrees less than kk might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored kk-Core} problem: Given a graph GG and integers b,kb, k and pp do there exist a set of vertices BβŠ†HβŠ†V(G)B\subseteq H\subseteq V(G) such that ∣Bβˆ£β‰€b,∣H∣β‰₯p|B|\leq b, |H|\geq p and every vertex v∈Hβˆ–Bv\in H\setminus B has degree at least kk is the induced subgraph G[H]G[H]. They showed that the problem is NP-hard for kβ‰₯2k\geq 2 and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the \textsc{Anchored kk-Core} problem is W[1]-hard parameterized by pp, even for k=3k=3. This improves the result of Bhawalkar et al. \cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by bb) as our parameter is always bigger since pβ‰₯bp\geq b. Then we answer a question of Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored kk-Core} problem remains NP-hard on planar graphs for all kβ‰₯3k\geq 3, even if the maximum degree of the graph is k+2k+2. Finally we show that the problem is FPT on planar graphs parameterized by bb for all kβ‰₯7k\geq 7.Comment: To appear in AAAI 201

    Approximating acyclicity parameters of sparse hypergraphs

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    The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph H is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class. This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class

    Refined Complexity of PCA with Outliers

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    Principal component analysis (PCA) is one of the most fundamental procedures in exploratory data analysis and is the basic step in applications ranging from quantitative finance and bioinformatics to image analysis and neuroscience. However, it is well-documented that the applicability of PCA in many real scenarios could be constrained by an "immune deficiency" to outliers such as corrupted observations. We consider the following algorithmic question about the PCA with outliers. For a set of nn points in Rd\mathbb{R}^{d}, how to learn a subset of points, say 1% of the total number of points, such that the remaining part of the points is best fit into some unknown rr-dimensional subspace? We provide a rigorous algorithmic analysis of the problem. We show that the problem is solvable in time nO(d2)n^{O(d^2)}. In particular, for constant dimension the problem is solvable in polynomial time. We complement the algorithmic result by the lower bound, showing that unless Exponential Time Hypothesis fails, in time f(d)no(d)f(d)n^{o(d)}, for any function ff of dd, it is impossible not only to solve the problem exactly but even to approximate it within a constant factor.Comment: To be presented at ICML 201
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