4,042 research outputs found

    Approximating Dense Max 2-CSPs

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    In this paper, we present a polynomial-time algorithm that approximates sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within O(Nε)O(N^{\varepsilon}) approximation ratio for any constant ε>0\varepsilon > 0. Using this algorithm, we also achieve similar results for free games, projection games on sufficiently dense random graphs, and the Densest kk-Subgraph problem with sufficiently dense optimal solution. Note, however, that algorithms with similar guarantees to the last algorithm were in fact discovered prior to our work by Feige et al. and Suzuki and Tokuyama. In addition, our idea for the above algorithms yields the following by-product: a quasi-polynomial time approximation scheme (QPTAS) for satisfiable dense Max 2-CSPs with better running time than the known algorithms

    Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

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    While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of {\em both} time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a problem which lies at the core of many graph mining applications called {\em densest subgraph problem}. We develop an algorithm that achieves time- and space-efficiency for this problem simultaneously. It is one of the first of its kind for graph problems to the best of our knowledge. In a graph G=(V,E)G = (V, E), the "density" of a subgraph induced by a subset of nodes SVS \subseteq V is defined as E(S)/S|E(S)|/|S|, where E(S)E(S) is the set of edges in EE with both endpoints in SS. In the densest subgraph problem, the goal is to find a subset of nodes that maximizes the density of the corresponding induced subgraph. For any ϵ>0\epsilon>0, we present a dynamic algorithm that, with high probability, maintains a (4+ϵ)(4+\epsilon)-approximation to the densest subgraph problem under a sequence of edge insertions and deletions in a graph with nn nodes. It uses O~(n)\tilde O(n) space, and has an amortized update time of O~(1)\tilde O(1) and a query time of O~(1)\tilde O(1). Here, O~\tilde O hides a O(\poly\log_{1+\epsilon} n) term. The approximation ratio can be improved to (2+ϵ)(2+\epsilon) at the cost of increasing the query time to O~(n)\tilde O(n). It can be extended to a (2+ϵ)(2+\epsilon)-approximation sublinear-time algorithm and a distributed-streaming algorithm. Our algorithm is the first streaming algorithm that can maintain the densest subgraph in {\em one pass}. The previously best algorithm in this setting required O(logn)O(\log n) passes [Bahmani, Kumar and Vassilvitskii, VLDB'12]. The space required by our algorithm is tight up to a polylogarithmic factor.Comment: A preliminary version of this paper appeared in STOC 201

    A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem

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    Many graph mining applications rely on detecting subgraphs which are near-cliques. There exists a dichotomy between the results in the existing work related to this problem: on the one hand the densest subgraph problem (DSP) which maximizes the average degree over all subgraphs is solvable in polynomial time but for many networks fails to find subgraphs which are near-cliques. On the other hand, formulations that are geared towards finding near-cliques are NP-hard and frequently inapproximable due to connections with the Maximum Clique problem. In this work, we propose a formulation which combines the best of both worlds: it is solvable in polynomial time and finds near-cliques when the DSP fails. Surprisingly, our formulation is a simple variation of the DSP. Specifically, we define the triangle densest subgraph problem (TDSP): given G(V,E)G(V,E), find a subset of vertices SS^* such that τ(S)=maxSVt(S)S\tau(S^*)=\max_{S \subseteq V} \frac{t(S)}{|S|}, where t(S)t(S) is the number of triangles induced by the set SS. We provide various exact and approximation algorithms which the solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to the more general problem of maximizing the kk-clique average density. Finally, we provide empirical evidence that the TDSP should be used whenever the output of the DSP fails to output a near-clique.Comment: 42 page

    Robust Densest Subgraph Discovery

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    Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph G=(V,E)G=(V,E) with an edge-weight vector w=(we)eEw=(w_e)_{e\in E}, we aim to find SVS\subseteq V that maximizes the density, i.e., w(S)/Sw(S)/|S|, where w(S)w(S) is the sum of the weights of the edges in the subgraph induced by SS. Although the densest subgraph problem is one of the most well-studied optimization problems for dense subgraph discovery, there is an implicit strong assumption; it is assumed that the weights of all the edges are known exactly as input. In real-world applications, there are often cases where we have only uncertain information of the edge weights. In this study, we provide a framework for dense subgraph discovery under the uncertainty of edge weights. Specifically, we address such an uncertainty issue using the theory of robust optimization. First, we formulate our fundamental problem, the robust densest subgraph problem, and present a simple algorithm. We then formulate the robust densest subgraph problem with sampling oracle that models dense subgraph discovery using an edge-weight sampling oracle, and present an algorithm with a strong theoretical performance guarantee. Computational experiments using both synthetic graphs and popular real-world graphs demonstrate the effectiveness of our proposed algorithms.Comment: 10 pages; Accepted to ICDM 201

    Core Decomposition in Multilayer Networks: Theory, Algorithms, and Applications

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    Multilayer networks are a powerful paradigm to model complex systems, where multiple relations occur between the same entities. Despite the keen interest in a variety of tasks, algorithms, and analyses in this type of network, the problem of extracting dense subgraphs has remained largely unexplored so far. In this work we study the problem of core decomposition of a multilayer network. The multilayer context is much challenging as no total order exists among multilayer cores; rather, they form a lattice whose size is exponential in the number of layers. In this setting we devise three algorithms which differ in the way they visit the core lattice and in their pruning techniques. We then move a step forward and study the problem of extracting the inner-most (also known as maximal) cores, i.e., the cores that are not dominated by any other core in terms of their core index in all the layers. Inner-most cores are typically orders of magnitude less than all the cores. Motivated by this, we devise an algorithm that effectively exploits the maximality property and extracts inner-most cores directly, without first computing a complete decomposition. Finally, we showcase the multilayer core-decomposition tool in a variety of scenarios and problems. We start by considering the problem of densest-subgraph extraction in multilayer networks. We introduce a definition of multilayer densest subgraph that trades-off between high density and number of layers in which the high density holds, and exploit multilayer core decomposition to approximate this problem with quality guarantees. As further applications, we show how to utilize multilayer core decomposition to speed-up the extraction of frequent cross-graph quasi-cliques and to generalize the community-search problem to the multilayer setting
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