Improving Local Search for Minimum Weighted Connected Dominating Set Problem by Inner-Layer Local Search

The minimum weighted connected dominating set (MWCDS) problem is an important variant of connected dominating set problems with wide applications, especially in heterogenous networks and gene regulatory networks. In the paper, we develop a nested local search algorithm called NestedLS for solving MWCDS on classic benchmarks and massive graphs. In this local search framework, we propose two novel ideas to make it effective by utilizing previous search information. First, we design the restart based smoothing mechanism as a diversification method to escape from local optimal. Second, we propose a novel inner-layer local search method to enlarge the candidate removal set, which can be modelled as an optimized version of spanning tree problem. Moreover, inner-layer local search method is a general method for maintaining the connectivity constraint when dealing with massive graphs. Experimental results show that NestedLS outperforms state-of-the-art meta-heuristic algorithms on most instances

On Approximating the Number of kk-cliques in Sublinear Time

We study the problem of approximating the number of $k$-cliques in a graph when given query access to the graph. We consider the standard query model for general graphs via (1) degree queries, (2) neighbor queries and (3) pair queries. Let $n$ denote the number of vertices in the graph, $m$ the number of edges, and $C_k$ the number of $k$-cliques. We design an algorithm that outputs a $(1+\varepsilon)$-approximation (with high probability) for $C_k$, whose expected query complexity and running time are O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log n,1/\varepsilon,k). Hence, the complexity of the algorithm is sublinear in the size of the graph for $C_k = \omega(m^{k/2-1})$. Furthermore, we prove a lower bound showing that the query complexity of our algorithm is essentially optimal (up to the dependence on $\log n$, $1/\varepsilon$ and $k$). The previous results in this vein are by Feige (SICOMP 06) and by Goldreich and Ron (RSA 08) for edge counting ($k=2$) and by Eden et al. (FOCS 2015) for triangle counting ($k=3$). Our result matches the complexities of these results. The previous result by Eden et al. hinges on a certain amortization technique that works only for triangle counting, and does not generalize for larger cliques. We obtain a general algorithm that works for any $k\geq 3$ by designing a procedure that samples each $k$-clique incident to a given set $S$ of vertices with approximately equal probability. The primary difficulty is in finding cliques incident to purely high-degree vertices, since random sampling within neighbors has a low success probability. This is achieved by an algorithm that samples uniform random high degree vertices and a careful tradeoff between estimating cliques incident purely to high-degree vertices and those that include a low-degree vertex

Fast Distributed Approximation for Max-Cut

Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $\epsilon > 0$, we develop randomized approximation algorithms achieving a ratio of $(1-\epsilon)$ to the optimum for Max-Cut on bipartite graphs in the $\mathcal{CONGEST}$ model, and on general graphs in the $\mathcal{LOCAL}$ model. We further present efficient deterministic algorithms, including a $1/3$-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of $1/4$. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest

The Power Of Locality In Network Algorithms

Over the last decade we have witnessed the rapid proliferation of large-scale complex networks, spanning many social, information and technological domains. While many of the tasks which users of such networks face are essentially global and involve the network as a whole, the size of these networks is huge and the information available to users is only local. In this dissertation we show that even when faced with stringent locality constraints, one can still effectively solve prominent algorithmic problems on such networks. In the first part of the dissertation we present a natural algorithmic framework designed to model the behaviour of an external agent trying to solve a network optimization problem with limited access to the network data. Our study focuses on local information algorithms --- sequential algorithms where the network topology is initially unknown and is revealed only within a local neighborhood of vertices that have been irrevocably added to the output set. We address both network coverage problems as well as network search problems. Our results include local information algorithms for coverage problems whose performance closely match the best possible even when information about network structure is unrestricted. We also demonstrate a sharp threshold on the level of visibility required: at a certain visibility level it is possible to design algorithms that nearly match the best approximation possible even with full access to the network structure, but with any less information it is impossible to achieve a reasonable approximation. For preferential attachment networks, we obtain polylogarithmic approximations to the problem of finding the smallest subgraph that connects a subset of nodes and the problem of finding the highest-degree nodes. This is achieved by addressing a decade-old open question of Bollobás and Riordan on locally finding the root in a preferential attachment process. In the second part of the dissertation we focus on designing highly time efficient local algorithms for central mining problems on complex networks that have been in the focus of the research community over a decade: finding a small set of influential nodes in the network, and fast ranking of nodes. Among our results is an essentially runtime-optimal local algorithm for the influence maximization problem in the standard independent cascades model of information diffusion and an essentially runtime-optimal local algorithm for the problem of returning all nodes with PageRank bigger than a given threshold. Our work demonstrates that locality is powerful enough to allow efficient solutions to many central algorithmic problems on complex networks