A Greedy Approximation for Minimum Cardinality Multiple Quasi-submodular Cover with Applications

The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multilayer networks, i.e., networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers and between activities and degrees. We show that real-world networks exhibit indeed nontrivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.

This chapter is devoted to the study of 16 types of greedy algorithms for decision tree construction. The dynamic programming approach is used for construction of optimal decision trees. Optimization is performed relative to minimal values of average depth, depth, number of nodes, number of terminal nodes, and number of nonterminal nodes of decision trees. We compare average depth, depth, number of nodes, number of terminal nodes and number of nonterminal nodes of constructed trees with minimum values of the considered parameters obtained based on a dynamic programming approach. We report experiments performed on data sets from UCI ML Repository and randomly generated binary decision tables. As a result, for depth, average depth, and number of nodes we propose a number of good heuristics.

Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical textsf {NP}-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta )-approximation for Minimum Vertex Cover and Minimum Dominating Set, and a Omega (Delta )-approximation for Maximum Independent Set. We prove that the PLB-U property with beta >2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds.

The Steiner tree problem is one of the fundamental optimization problems in distributed graph algorithms. Recently Saikia and Karmakar [27] proposed a deterministic distributed algorithm for the Steiner tree problem that constructs a Steiner tree in \(O(S + \sqrt{n} \log ^* n)\) rounds whose cost is optimal upto a factor of \(2(1 - 1/\ell )\), where n and S are the number of nodes and shortest path diameter [17] respectively of the given input graph and \(\ell \) is the number of terminal leaf nodes in the optimal Steiner tree. The message complexity of the algorithm is \(O(\varDelta (n - t) S + n^{3/2})\), which is equivalent to \(O(mS + n^{3/2})\), where \(\varDelta \) is the maximum degree of a vertex in the graph, t is the number of terminal nodes (we assume that \(t < n\)), and m is the number of edges in the given input graph. This algorithm has a better round complexity than the previous best algorithm for Steiner tree construction due to Lenzen and Patt-Shamir [21]. In this paper we present a deterministic distributed algorithm which constructs a Steiner tree in \(\tilde{O}(S + \sqrt{n})\) rounds and \(\tilde{O}(mS)\) messages and still achieves an approximation factor of \(2(1 - 1/\ell )\). Note here that \(\tilde{O}(\cdot )\) notation hides polylogarithmic factors in n. This algorithm improves the message complexity of Saikia and Karmakar’s algorithm by dropping the additive term of \(O(n^{3/2})\) at the expense of a logarithmic multiplicative factor in the round complexity. Furthermore, we show that for sufficiently small values of the shortest path diameter \((S=O(\log n))\), a \(2(1 - 1/\ell )\)-approximate Steiner tree can be computed in \(\tilde{O}(\sqrt{n})\) rounds and \(\tilde{O}(m)\) messages and these complexities almost coincide with the results of some of the singularly-optimal minimum spanning tree (MST) algorithms proposed in [9, 12, 23].

Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In contrast, only few analytical results are known. We show that MinGreedy cannot improve on the trivial approximation ratio of $$\frac{1}{2}$$12 whp., even for bipartite graphs. Our hard inputs seem to require a small number of high-degree nodes. This motivates an investigation of greedy algorithms on graphs with maximum degree $$\varDelta $$Δ: we show that MinGreedy achieves a $${\frac{{\varDelta }-1}{2{\varDelta }-3}} $$Δ-12Δ-3-approximation for graphs with $${\varDelta } {=} 3$$Δ=3 and for $$\varDelta $$Δ-regular graphs, and a guarantee of $${\frac{{\varDelta }-1/2}{2{\varDelta }-2}} $$Δ-1/22Δ-2 for graphs with maximum degree $${\varDelta } $$Δ. Interestingly, our bounds even hold for the deterministic MinGreedy that breaks all ties arbitrarily. Moreover, we investigate the limitations of the greedy paradigm, using the model of priority algorithms introduced by Borodin, Nielsen, and Rackoff. We study deterministic priority algorithms and prove a $${\frac{{\varDelta }-1}{2{\varDelta }-3}}$$Δ-12Δ-3-inapproximability result for graphs with maximum degree $${\varDelta } $$Δ; thus, these greedy algorithms do not achieve a $$\frac{1}{2} {+} \varepsilon $$12+?-approximation and in particular the $$\frac{2}{3}$$23-approximation obtained by the deterministic MinGreedy for $${\varDelta } {=} 3$$Δ=3 is optimal in this class. For k-uniform hypergraphs we show a tight $$\frac{1}{k}$$1k-inapproximability bound. We also study fully randomized priority algorithms and give a $$\frac{5}{6}$$56-inapproximability bound. Thus, they cannot compete with matching algorithms of other paradigms.

In this paper, we present two techniques to analyze greedy approximation with nonsubmodular functions restricted submodularity and shifted submodularity. As an application of the restricted submodularity, we present a worst-case analysis of a greedy algorithm for Network Steiner tree adapted from a heuristic originally proposed by Chang in 1972 for Euclidean Steiner tree. The performance ratio of Chang's heuristic is a longstanding open problem due to the nonsubmodularity of its potential function. As an application of the shifted submodularity, we present a worst-case analysis of a greedy algorithm for Connected Dominating Set generalized from a greedy algorithm proposed by Ruan et al. Such generalized greedy algorithm is shown to have performance ratio at most (1 + e)(1 + ln(Δ - 1)), which matches the well-known lower bound (1-e)ln Δ, where Δ is the maximum vertex-degree of input graph and e is any positive constant.

In this paper, we study the design of fault tolerant networks for arrays and meshes by adding redundant nodes and edges. For a target graph G (linear array or mesh in this paper), a graph G′ is called a k-fault-tolerant graph of G if when we remove any k nodes from G′, it still contains a subgraph isomorphic to G. The major quality measures for a fault-tolerant graph are the number of spare nodes it uses and the maximum degree it has. The degree is particularly important in practice as it poses constraints on the scalability of the system. In this paper, we aim at designing fault-tolerant graphs with both small degree and small number of spare nodes. The graphs we obtain have degree O(1) for arrays and O(log3 k) for meshes. The number of spare nodes used are O(k log2 k) and O(k2/log k), respectively. Compared to the previous results, the number of spare nodes used in our construction has one fewer linear factor in k.

Local search for weighted sum coloring problem

PACK: Path coloring based k-connectivity detection algorithm for wireless sensor networks

The general theory of greedy approximation with respect to arbitrary dictionaries is well developed in the case of real Banach spaces. Recently some results proved for the Weak Chebyshev Greedy Algorithm (WCGA) in the case of real Banach spaces were extended to the case of complex Banach spaces. In this paper we extend some of the results known in the real case for greedy algorithms other than the WCGA to the case of complex Banach spaces. Bibliography: 25 titles.

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p^{*}. This transition is such that dynamical processes are separated into two regimes: if p>p^{*}, the network acts as a whole; whereas when p<p^{*}, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p^{*} to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p^{*} in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p^{*} in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

Quantum transport through disordered structures is inhibited by localized eigenstates of the Hamiltonian associated with the network. We show how to overcome localization by network multiplexing. Here, multiple layers of random networks with the same number of nodes are stacked in such a way that in the perpendicular directions regular one-dimensional networks are formed. Depending on the ratio of the coupling within the layer and perpendicular to it, transport gets either enhanced or diminished. In particular, if the couplings are of the same order, transport gets enhanced and localization effects can be overcome. We exemplify our results by multiplexes of random networks, where the disorder is topological.

In the minimum entropy set cover problem, one is given a collection of k sets which collectively cover an n-element ground set. A feasible solution of the problem is a partition of the ground set into parts such that each part is included in some of the k given sets. Such a partition defines a probability distribution, obtained by dividing each part size by n. The goal is to find a feasible solution minimizing the (binary) entropy of the corresponding distribution. Halperin and Karp have recently proved that the greedy algorithm always returns a solution whose cost is at most the optimum plus a constant. We improve their result by showing that the greedy algorithm approximates the minimum entropy set cover problem within an additive error of 1 nat =log 2 e bits $\simeq$1.4427 bits. Moreover, inspired by recent work by Feige, Lovasz and Tetali on the minimum sum set cover problem, we prove that no polynomial-time algorithm can achieve a better constant, unless P = NP. We also discuss some consequences for the related minimum entropy coloring problem.

International audience

Sparse wavelength conversion can increase the performance of all-optical wavelength division multiplexing (WDM) networks signi cantly by relaxing the wavelength continuity constraint. In this paper, we study the wavelength converter placement problem in multi- ber networks with static traf c demands. We present a tabu search based heuristic algorithm. The objective of the algorithm is to satisfy all the traf c demands with the minimum total cost of bers achieved in the full conversion case, by placing minimum number of wavelength converting nodes. We also implement a greedy algorithm and compare the performances of these converter placement algorithms with the optimum solutions on a sample network. The Tabu search based algorithm achieves the optimum solution in 72% of the test cases and it increases the average number of wavelength converting nodes by less than 10% with respect to the optimum solution. The effect of the utilized routing scheme on the generated solutions and the correlation between the converter node locations and the amount of traf c passing through the nodes are also investigated.