We develop efficient and effective strategies for the update of Katz centralities after node and edge removal in simple graphs. We provide explicit formulas for the "loss of walks" a network suffers when nod...
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The dual of a planar graph G is a planar graph G∗ that has a vertex for each face of G and an edge for each pair of adjacent faces of G. The profound relationship between a planar graph and its dual has been the algor...
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The dual of a planar graph G is a planar graph G∗ that has a vertex for each face of G and an edge for each pair of adjacent faces of G. The profound relationship between a planar graph and its dual has been the algorithmic basis for solving numerous (centralized) classical problems on planar graphs involving distances, flows, and cuts. In the distributed setting however, the only use of planar duality is for finding a recursive decomposition of G [DISC 2017, STOC 2019]. In this paper, we initiate the study of distributed algorithms on dual planar graphs. Namely, we extend the distributed algorithmic toolkit (such as recursive decomposition and minor-aggregation) to work on the dual graph G∗. These tools can then facilitate various algorithms on G by solving a suitable dual problem on G∗. Given a directed planar graph G with positive and negative edge-lengths and hop-diameter D, our key result is an Õ(D2)-round algorithm1 for Single Source Shortest Paths on G∗, which then implies an Õ(D2)-round algorithm for Maximum st-Flow on G. Prior to our work, no Õ(poly(D))-round algorithm was known for Maximum st-Flow. We further obtain a D · no(1)-rounds (1 − ∊)-approximation algorithm for Maximum st-Flow on G when G is undirected and s and t lie on the same face. Finally, we give a near optimal Õ(D)-round algorithm for computing the weighted girth of G. We believe that the toolkit developed in this paper for exploiting planar duality will be used in future distributed algorithms for various other classical problems on planar graphs (as happened in the centralized setting). The main challenges in our work are that G∗ is not the communication graph (e.g., a vertex of G is mapped to multiple vertices of G∗), and that the diameter of G∗ can be much larger than D (i.e., possibly by a linear factor). We overcome these challenges by carefully defining and maintaining subgraphs of the dual graph G∗ while applying the recursive decomposition on the primal graph G. The main technical
In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed gr...
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Optimizing data movements during program executions is essential for achieving high performance in modern computing systems. This has been classically modeled with the Red-Blue Pebble Game and its variants. In the exi...
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In this paper, we present an algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks. We first propose a novel model, in which we integrate the Preferential Attachment ...
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In this paper, we present an algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks. We first propose a novel model, in which we integrate the Preferential Attachment (PA) model for popularity growth with the Independent Cascade (IC) model for influence propagation in social networks called PA-IC model. In PA-IC, a popular item and a novice item grab shares of popularity from the natural popularity growth via the PA model, while the novice item tries to gain extra popularity via influence cascade in a social network. The popularity ratio is defined as the ratio of the popularity measure between the novice item and the popular item. We formulate Popularity Ratio Maximization (PRM) as the problem of selecting seeds in multiple rounds to maximize the popularity ratio in the end. We analyze the popularity ratio and show that it is monotone but not submodular. To provide an effective solution, we devise a surrogate objective function and show that empirically it is very close to the original objective function while theoretically, it is monotone and submodular. We design two efficient algorithms, one for the overlapping influence and non-overlapping seeds (across rounds) setting and the other for the non-overlapping influence and overlapping seed setting, and further discuss how to deal with other models and problem variants. Our empirical evaluation further demonstrates that our proposed method consistently achieves the best popularity promotion compared to other methods. Our theoretical and empirical analyses shed light on the interplay between influence maximization and preferential attachment in social networks.
Given an undirected graph G = (V, E), vertices s, t E V, and an integer k, TRACKING SHORTEST PATHS requires deciding whether there exists a set of k vertices T c_ V such that for any two distinct shortest paths betwee...
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Given an undirected graph G = (V, E), vertices s, t E V, and an integer k, TRACKING SHORTEST PATHS requires deciding whether there exists a set of k vertices T c_ V such that for any two distinct shortest paths between s and t, say P1 and P2, we have T n V(P1) =? T n V(P2). In this paper, we give the first polynomial size kernel for the problem. Specifically we show the existence of a kernel with O(k2) vertices and edges in general graphs and a kernel with O(k) vertices and edges in planar graphs for the TRACKING PATHS IN DAG problem. This problem admits a polynomial parameter transformation to TRACKING SHORTEST PATHS, and this implies a kernel with O(k4) vertices and edges for TRACKING SHORTEST PATHS in general graphs and a kernel with O(k2) vertices and edges in planar graphs. Based on the above we also give a single exponential algorithm for TRACKING SHORTEST PATHS in planar graphs. (c) 2022 Elsevier B.V. All rights reserved.
In the CACTUS VERTEX DELETION (resp., EVEN CYCLE TRANSVERSAL) problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G results in a ...
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In the CACTUS VERTEX DELETION (resp., EVEN CYCLE TRANSVERSAL) problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G results in a graph in which every edge belongs to at most one cycle (resp., a graph without even cycles). In this paper we give deterministic O *(13.69k)-time algorithms for CACTUS VERTEX DELETION and EVEN CYCLE TRANSVERSAL.(c) 2022 Elsevier B.V. All rights reserved.
In the estimation of causal effects, one common method for removing the influence of confounders is to adjust the variables that satisfy the back-door criterion. However, it is not always possible to uniquely determin...
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The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of ...
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A graph property is elusive (or evasive) if any algorithm testing it by asking questions of the form "Is there an edge between vertices x and y?" must, in the worst case, examine all pairs of vertices. Elusi...
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