The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

Lucibello, Carlo; Ricci-Tersenghi, Federico

doi:10.1155/2014/136829

Cited by 13 publications

(14 citation statements)

References 32 publications

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“…(23), (24), (25) (or Eq. (29), (30), (31)) are very difficult to solve, because every edge direction has three variables and three associated equations. Therefore, in order to avoid this high complexity, we use a coarse-grained method.…”

Section: Cavity Methods Analysismentioning

confidence: 99%

Analytical solution for the size of the minimum dominating set in complex networks

Nacher

Ochiai

2017

Int. J. Model. Simul. Sci. Comput.

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Domination is the fastest-growing field within graph theory with a profound diversity and impact in real-world applications, such as the recent breakthrough approach that identifies optimized subsets of proteins enriched with cancer-related genes. Despite its conceptual simplicity, domination is a classical NP-complete decision problem which makes analytical solutions elusive and poses difficulties to design optimization algorithms for finding a dominating set of minimum cardinality in a large network. Here we derive for the first time an approximate analytical solution for the density of the minimum dominating set (MDS) by using a combination of cavity method and Ultra-Discretization (UD) procedure. The derived equation allows us to compute the size of MDS by only using as an input the information of the degree distribution of a given network.The research on complex networks in diverse fields [1,2], based on applied graph theory combined with computational and statistical physics methods, has experienced a spectacular growth in recent years and has led to the discovery of ubiquitous patterns called scale-free networks [3], unexpected dynamic behavior [4], robustness and vulnerability features [5-7], and applications in natural and social complex systems [1, 2]. On the other hand, domination is an important problem in graph theory which has rich variants, such as independence, covering and matching [8]. The mathematical and computational studies on domination have led to abundant applications in disparate fields such as mobile computing and computer communication networks [9], design of large parallel and distributed systems [10], analysis of large social networks [11-13], computational biology and biomedical analysis [14] and discrete algorithms research [8].More recently, the Minimum Dominating Set (MDS) has drawn the attention researchers to controllability in complex networks [15][16][17], to investigate observability in power-grid [18] and to identify an optimized subset of proteins enriched with essential, cancer-related and virus-targeted genes in protein networks [19]. The size of MDS was also investigated by extensively analysing several types of artificial scale-free networks using a greedy algorithm in [20]. The problem to design complex networks that are structurally robust has also recently been investigated using the MDS approach [21,22]. Despite its conceptual simplicity, the MDS is a classical NP-complete decision problem in computational complexity theory [23]. Therefore, it is believed that there is no a theoretically efficient (i.e., polynomial time) algorithm that finds an exact smallest dominating set for a given graph. It is worth noticing that although it is an NP-hard problem, recent results have shown that we can use Integer Linear Programming (ILP) to find optimal solution [15,19,21]. Moreover, for specific types of graph such a tree (i.e., no loops) and even partial-k-tree, it can be solved using Dynamic Programing (DP) in polynomial time [24].Especially, the density (or fraction) of MD...

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Section: Cavity Methods Analysismentioning

confidence: 99%

Analytical solution for the size of the minimum dominating set in complex networks

Nacher

Ochiai

2017

Int. J. Model. Simul. Sci. Comput.

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“…The second step is to write ρ α as a function of the degree distribution of G D . We do this by building on recent work by Lucibello and Ricci-Tersenghi [26], we calculated an estimation for ρ α based on a factor graph description of the maximum set packing problem, of which the independence number problem is a particular instance. Crucially, these expressions depend only on the graph's degree distribution, which in turn allows to study the effect that different network topologies have on their parallel capacity to be studied.…”

Section: B Maximum Parallel Capacity Estimation For Dependency Graphs...mentioning

confidence: 99%

Topological limits to parallel processing capability of network architectures

Petri¹,

Musslick²,

Dey³

et al. 2017

Preprint

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The ability to learn new tasks and generalize performance to others is one of the most remarkable characteristics of the human brain and of recent AI systems. The ability to perform multiple tasks simultaneously is also a signature characteristic of large-scale parallel architectures, that is evident in the human brain, and has been exploited effectively in more traditional, massively parallel computational architectures. Here, we show that these two characteristics are in tension, reflecting a fundamental tradeoff between interactive parallelism that supports learning and generalization, and independent parallelism that supports processing efficiency through concurrent multitasking. We formally show that, while the maximum number of tasks that can be performed simultaneously grows linearly with network size, under realistic scenarios (e.g. in an unpredictable environment), the expected number that can be performed concurrently grows radically sub-linearly with network size. Hence, even modest reliance on shared representation strictly constrains the number of tasks that can be performed simultaneously, implying profound consequences for the development of artificial intelligence that optimally manages the tradeoff between learning and processing, and for understanding the human brains remarkably puzzling mix of sequential and parallel capabilities.

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“…Here we extend the leaf-removal idea of [17] (see also more recent work [6,18,19]) and consider a generalized leaf-removal process. This dynamics is based on the following two considerations: first, as…”

Section: The Glr Processmentioning

confidence: 99%

Statistical Mechanics of the Minimum Dominating Set Problem

2015

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The minimum dominating set (MDS) problem has wide applications in network science and related fields. It aims at constructing a node set of smallest size such that any node of the network is either in this set or is adjacent to at least one node of this set. Although this optimization problem is generally very difficult, we show it can be exactly solved by a generalized leaf-removal (GLR) process if the network contains no core. We present a percolation theory to describe the GLR process on random networks, and solve a spin glass model by mean field method to estimate the MDS size. We also implement a message-passing algorithm and a local heuristic algorithm that combines GLR with greedy node-removal to obtain near-optimal solutions for single random networks. Our algorithms also perform well on real-world network instances.

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The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

Cited by 13 publications

References 32 publications

Analytical solution for the size of the minimum dominating set in complex networks

Analytical solution for the size of the minimum dominating set in complex networks

Topological limits to parallel processing capability of network architectures

Statistical Mechanics of the Minimum Dominating Set Problem

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