Clique Model Research Articles

Analysis of Correlation Effects of Double Mutations in Enzymes: A Revised Residual-Contact Network Clique Model.

The relationship between amino acid mutations and enzyme bioactivity is a significant challenge in modern bio-industrial applications. Despite many successful designs relying on complex correlations among mutations at different enzyme sites, the underlying mechanisms of these correlations still need to be explored. In this study, we introduced a revised version of the residual-contact network clique model to investigate the additive effect of double mutations based on the mutation occurrence topology, secondary structures, and physicochemical properties. The model was applied to a set of 182 double mutations reported in three extensively studied enzymes, and it successfully identified over 90% of additive double mutations and a majority of non-additive double mutations. The calculations revealed that the mutation additivity depends intensely on the studied mutation sites' topology and physicochemical properties. For example, double mutations on irregular secondary structure regions tend to be non-additive. Our method provides valuable tools for facilitating enzyme design and optimization. The code and relevant data are available at Github.

Maximum k-Plex Computation: Theory and Practice

The k-plex model relaxes the clique model by allowing each vertex to miss up to k neighbors, including the vertex itself. A 1-plex is a clique. Many exact algorithms have been recently designed for finding the k-plex with the largest number of vertices, known as the maximum k-plex computation problem. However, all the existing algorithms, except BS, has the trivial worst-case time complexity of O*(2n) when ignoring polynomial factors. On the other hand, although BS improves the time complexity to O*(βkn) where βk < 2 is a constant depending only on k, its practical performance is not satisfactory. In this paper, we study the maximum k-plex computation problem from both theory and practice. We first propose two new reduction rules and a new branching rule and prove that the base of the exponential time complexity is reduced to γk when the new reduction and branching rules are incorporated into a standard backtracking algorithm; here γk < βk. We then design a two-stage approach kPlexT to improve the exponent of the time complexity by separating the search of large k-plexes from the search of small ones. We prove that kPlexT runs in O*((α Δ)k+1 γ_kα) time when the maximum k-plex size Ωk(G) is at least 2k-1, and in O*((α Δ)k+1 γ_kα + min(γkn, n2k-2)) time otherwise; here, α is the degeneracy and Δ is the maximum degree of the input graph. We also prove that with slight modification, kPlexT runs in O*((αΔ)k+1 (k+1)α+k-Ωk(G)) time when ømega_k(G) ≥ 2k-1. Finally, we propose another reduction rule and a better initialization method to improve the practical performance of kPlexT. Extensive empirical studies demonstrate that kPlexT achieves state-of-the-art practical performance. We also show that our improved time complexity carries over to other related problems such as enumerating all maximal k-plexes, quasi-cliques, and k-biplexes.

An optimization algorithm for maximum quasi-clique problem based on information feedback model.

The maximum clique problem in graph theory is a well-known challenge that involves identifying the complete subgraph with the highest number of nodes in a given graph, which is a problem that is hard for nondeterministic polynomial time (NP-hard problem). While finding the exact application of the maximum clique problem in the real world is difficult, the relaxed clique model quasi-clique has emerged and is widely applied in fields such as bioinformatics and social network analysis. This study focuses on the maximum quasi-clique problem and introduces two algorithms, NF1 and NR1. These algorithms make use of previous iteration information through an information feedback model, calculate the information feedback score using fitness weighting, and update individuals in the current iteration based on the benchmark algorithm and selected previous individuals. The experimental results from a significant number of composite and real-world graphs indicate that both algorithms outperform the original benchmark algorithm in dense instances, while also achieving comparable results in sparse instances.

Balanced Clique Computation in Signed Networks: Concepts and Algorithms

Clique is one of the most fundamental models for cohesive subgraph mining in network analysis. Existing clique model mainly focuses on unsigned networks. However, in real world, many applications are modeled as signed networks with positive and negative edges. As the signed networks hold their own properties different from the unsigned networks, the existing clique model is inapplicable for the signed networks. Motivated by this, we propose the balanced clique model that considers the most fundamental and dominant theory, structural balance theory, for signed networks. Following the balanced clique model, we study the <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</u> aximal <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</u> alanced <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</u> lique <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</u> numeration problem (MBCE) which computes all the maximal balanced cliques in a given signed network. Moreover, in some applications, users prefer a unique and representative balanced clique with maximum size rather than all balanced cliques. Thus, we also study the <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</u> aximum <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</u> alanced <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</u> lique <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</u> earch problem (MBCS) which computes the balanced clique with maximum size. We show that MBCE problem and MBCS problem are both NP-Hard. For the MBCE problem, a straightforward solution is to treat the signed network as two unsigned networks and leverage the off-the-shelf techniques for unsigned networks. However, such a solution is inefficient for large signed networks. To address this problem, in this paper, we first propose a new maximal balanced clique enumeration algorithm by exploiting the unique properties of signed networks. Based on the new proposed algorithm, we devise two optimization strategies to further improve the efficiency of the enumeration. For the MBCS problem, we first propose a baseline solution. To overcome the huge search space problem of the baseline solution, we propose a new search framework based on search space partition. To further improve the efficiency of the new framework, we propose multiple optimization strategies regarding to redundant search branches and invalid candidates. We conduct extensive experiments on large real datasets. The experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms for MBCE problem and MBCS problem.

Fairness-Aware Maximal Clique in Large Graphs: Concepts and Algorithms

Cohesive subgraph mining on attributed graphs is a fundamental problem in graph data analysis. Existing cohesive subgraph mining algorithms on attributed graphs do not consider the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fairness</i> of attributes in the subgraph. In this paper, we, for the first time, introduce fairness into the widely-used clique model to mine fairness-aware cohesive subgraphs. In particular, we propose three novel fairness-aware maximal clique models on attributed graphs, called weak fair clique, strong fair clique and relative fair clique, respectively. To enumerate all weak fair cliques, we develop an efficient backtracking algorithm called WFCEnum equipped with a novel colorful <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core based pruning technique. We also propose an efficient enumeration algorithm called SFCEnum to find all strong fair cliques based on a new attribute-alternatively-selection search technique. To further improve the efficiency, we also present several non-trivial ordering techniques for both weak and strong fair clique enumerations. To enumerate all relative fair cliques, we design an enhanced colorful <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -core based pruning technique for 2D attributes, and develop two efficient search algorithms: RFCRefineEnum and RFCAlterEnum for arbitrary dimension attributes. The results of extensive experiments on four real-world graphs demonstrate the efficiency, scalability and effectiveness of the proposed algorithms.

Almost universally optimal distributed Laplacian solvers via low-congestion shortcuts

In this paper, we refine the (almost) existentially optimal distributed Laplacian solver of Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS ‘21) into an (almost) universally optimal distributed Laplacian solver. Specifically, when the topology is known (i.e., the Supported-CONGEST model), we show that any Laplacian system on an n-node graph with shortcut qualitytextrm{SQ}(G) can be solved after n^{o(1)} text {SQ}(G) log (1/epsilon ) rounds, where epsilon >0 is the required accuracy. This almost matches our lower bound that guarantees that any correct algorithm on G requires widetilde{Omega }(textrm{SQ}(G)) rounds, even for a crude solution with epsilon le 1/2. Several important implications hold in the unknown-topology (i.e., standard CONGEST) case: for excluded-minor graphs we get an almost universally optimal algorithm that terminates in D cdot n^{o(1)} log (1/epsilon ) rounds, where D is the hop-diameter of the network; as well as n^{o(1)} log (1/epsilon )-round algorithms for the case of textrm{SQ}(G) le n^{o(1)}, which holds for most networks of interest. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated clique model. In this model, we show the existence of a Laplacian solver with round complexity n^{o(1)} log (1/epsilon ). The unifying thread of these results, and our main technical contribution, is the development of near-optimal algorithms for a novel rho -congested generalization of the standard part-wise aggregation problem, which could be of independent interest.

Research on variable weight CLIQUE clustering algorithm based on partial order set1

The uncertainty of weight makes the weight density between samples not fixed. Aiming at the problem that the existing CLIQUE clustering algorithm does not consider the weight of object features, which leads to low accuracy, an improved weighted method combined with the thought of posets is proposed. In addition, this method does not need accurate weight assignment, only the weight order can run efficiently. First, the weight order of object features is obtained, and then the partial order weight is applied to the original data to obtain weighted data with weights. Then the traditional CLIQUE algorithm is used to cluster according to weighted data, and finally the partial order weighted CLIQUE model is obtained. Through the experiment of six groups of data, the results show that: under the given weight sequence constraints, the clustering quality of the weighted CLIQUE model is significantly higher than that of the unweighted model, and the clustering accuracy and other aspects are significantly improved. In this method model, weight information is effectively integrated into the algorithm when only the feature weight order is obtained, and the function of feature weight is fully played to enhance the robustness of clustering results. At the same time, the idea of poset can effectively integrate expert information, and the representation of the nearest neighbor elements in Hasse graph can show the effect intuitively. It is an effective improvement method of CLIQUE clustering algorithm.

Maximal Defective Clique Enumeration

Maximal clique enumeration is a fundamental operator in graph analysis. The model of clique, however, is typically too restrictive for real-world applications as it requires an edge for every pair of vertices. To remedy this restriction, practical graph analysis applications often resort to find relaxed cliques as alternatives. In this work, we investigate a notable relaxed clique model, called s-defective clique, which allows at most s edges to be missing. Similar to the complexity of maximal clique enumeration, the problem of enumerating all maximal s-defective cliques is also NP-hard. To solve this problem, we first develop a new polynomial-delay algorithm based on a carefully-designed reverse search technique, which can output two consecutive results within polynomial time. To achieve better practical efficiency, we propose a branch-and-bound algorithm with a novel pivoting technique. We prove that the time complexity of this algorithm depends only on O(α_sn) or O(αsδ) when using a degeneracy ordering optimization, where αs is a positive real number strictly less than 2, and δ (δ &lt;n) is the degeneracy of the graph. To our knowledge, this is the first algorithm that can break the O(2n) time complexity to enumerate all maximal s-defective cliques (s&gt;0). We also develop several new pruning techniques to further improve the efficiency of our branch-and-bound algorithm to enumerate all relatively-large maximal s-defective cliques. In addition, we further generalize our pivot-based branch-and-bound algorithm to enumerate all maximal subgraphs satisfying a hereditary property. Here we call a graph meeting the hereditary property if all its subgraphs have the same property as itself. Finally, extensive experiments on 11 datasets demonstrate the efficiency, effectiveness, and scalability of the proposed solutions.

Efficient maximum k -plex computation over large sparse graphs

The k -plex model is a relaxation of the clique model by allowing every vertex to miss up to k neighbors. Designing exact and efficient algorithms for computing a maximum k -plex in a graph has been receiving increasing interest recently. However, the existing algorithms are still inefficient due to having major limitations. We in this paper design a new algorithm kPlexS for the maximum k -plex problem, with three novel contributions. Firstly, we propose a new framework for computing maximum k -plex over large sparse graphs, by iteratively extracting small dense subgraphs from it and then solving each of the extracted dense subgraphs by a branch-and-bound search. Secondly, we propose an efficient reduction algorithm CTCP to reduce the input graph size by exhaustively conducting vertex reduction and edge reduction. CTCP computes a smaller reduced graph and also has a lower time complexity than the existing techniques. Moreover, we iteratively invoke CTCP to reduce the input graph once a vertex has been processed and removed from it. Thirdly, we develop a branch-and-bound algorithm BBMatrix specifically targeting the dense subgraphs that are extracted from the input graph. BBMatrix represents its input graph by an adjacency matrix, and utilizes both first-order (i.e., individual vertices) and second-order information (i.e., pairs of vertices) for reduction and upper bounding. In addition, incremental techniques are proposed to efficiently apply the reduction and upper bounding during the recursion. Extensive empirical studies on large real graphs demonstrate that our algorithm kPlexS outperforms the state-of-the-art algorithms BnB, Maplex, and KpLeX.

Distributed Graph Realizations

We study graph realization problems for the first time from a distributed perspective. Graph realization problems are encountered in distributed construction of overlay networks that must satisfy certain degree or connectivity properties. We study them in the node capacitated clique (NCC) model of distributed computing, recently introduced for representing peer-to-peer overlay networks. We focus on two central variants, <i>degree-sequence realization</i> and <i>minimum threshold-connectivity</i> realization. In the degree sequence problem, each node <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> is associated with a degree <inline-formula><tex-math notation="LaTeX">$d(v)$</tex-math></inline-formula> , and the resulting degree sequence is realizable if it is possible to construct an overlay network in which the degree of each node <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> is <inline-formula><tex-math notation="LaTeX">$d(v)$</tex-math></inline-formula> . The minimum threshold-connectivity problem requires us to construct an overlay network that satisfies connectivity constraints specified between every pair of nodes. Overlay network realizations can be either explicit or implicit. Explicit realizations require both endpoints of any edge in the realized graph to be aware of the edge. In implicit realizations, on the other hand, at least one endpoint of each edge of the realized graph needs to be aware of the edge. The main realization algorithms we present are the following. (Note that all our algorithms are randomized Las Vegas algorithms unless specified otherwise. The stated running times hold with high probability.) 1) An <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\min \lbrace \sqrt{m},\Delta \rbrace)$</tex-math></inline-formula> time algorithm for implicit realization of a degree sequence. Here, <inline-formula><tex-math notation="LaTeX">$\Delta = \max _v d(v)$</tex-math></inline-formula> is the maximum degree and <inline-formula><tex-math notation="LaTeX">$m = (1/2) \sum _v d(v)$</tex-math></inline-formula> is the number of edges in the final realization. 2) <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\Delta)$</tex-math></inline-formula> time algorithm for an explicit realization of a degree sequence. We first compute an implicit realization and then transform it into an explicit one in <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\Delta)$</tex-math></inline-formula> additional rounds. 3) An <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\Delta)$</tex-math></inline-formula> time algorithm for the threshold connectivity problem that obtains an explicit solution and an improved <inline-formula><tex-math notation="LaTeX">$\tilde{O}(1)$</tex-math></inline-formula> algorithm for implicit realization when all nodes know each other’s IDs. These algorithms yield 2-approximations w.r.t. the number of edges. We complement our upper bounds with lower bounds to show that the above algorithms are tight up to factors of <inline-formula><tex-math notation="LaTeX">$\log n$</tex-math></inline-formula> . Additionally, we provide algorithms for realizing trees (including a procedure for obtaining a tree with a minimal diameter), an <inline-formula><tex-math notation="LaTeX">$\tilde{O}(1)$</tex-math></inline-formula> round algorithm for approximate degree sequence realization and finally an <inline-formula><tex-math notation="LaTeX">$O(\log ^2 n)$</tex-math></inline-formula> algorithm for degree sequence realization in the non-preassigned case namely, where the input degree sequence may be permuted among the nodes.

Proof-labeling schemes: Broadcast, unicast and in between

We study the effect of limiting the number of different messages a node can transmit simultaneously on the verification complexity of proof-labeling schemes (PLS). In a PLS, each node is given a label, and the goal is to verify, by exchanging messages over each link in each direction, that a certain global predicate is satisfied by the system configuration. We consider a single parameter r that bounds the number of distinct messages that can be sent concurrently by any node: in the case r=1, each node may only send the same message to all its neighbors (the broadcast model), in the case r≥Δ, where Δ is the largest node degree in the system, each neighbor may be sent a distinct message (the unicast model), and in general, for 1≤r≤Δ, each of the r messages is destined to a subset of the neighbors.We show that message compression linear in r is possible for verifying fundamental problems such as the agreement between edge endpoints on the edge state. Some problems, including verification of maximal matching, exhibit a large gap in complexity between r=1 and r>1. For some other important predicates, the verification complexity is insensitive to r, e.g., the question whether a subset of edges constitutes a spanning-tree. We also consider the congested clique model. We show that the crossing technique [1] for proving lower bounds on the verification complexity can be applied in the case of congested clique only if r=1. Together with a new upper bound, this allows us to determine the verification complexity of MST in the broadcast clique. Finally, we establish a general connection between the deterministic and randomized verification complexity for any given number r.

A Planted Clique Perspective on Hypothesis Pruning

Hypothesis pruning is an important prerequisite while working with outlier-contaminated data in many computer vision problems. However, the underlying random data structures are barely explored in the literature, limiting designing efficient algorithms. To this end, we provide a novel graph-theoretic perspective on hypothesis pruning exploiting invariant structures of data. We introduce the planted clique model, a central object in computational statistics, to investigate the information-theoretical and computational limits of the hypothesis pruning problem. In addition, we propose an inductive learning framework for finding hidden cliques that learns heuristics on synthetic graphs with planted cliques and generalizes to real vision problems. We present competitive experimental results with large runtime improvement on synthetic and widely used vision datasets to show its efficacy.

Mismatching as a tool to enhance algorithmic performances of Monte Carlo methods for the planted clique model

Over-parametrization was a crucial ingredient for recent developments in inference and machine-learning fields. However a good theory explaining this success is still lacking. In this paper we study a very simple case of mismatched over-parametrized algorithm applied to one of the most studied inference problem: the planted clique problem. We analyze a Monte Carlo (MC) algorithm in the same class of the famous Jerrum algorithm. We show how this MC algorithm is in general suboptimal for the recovery of the planted clique. We show however how to enhance its performances by adding a (mismatched) parameter: the temperature; we numerically find that this over-parametrized version of the algorithm can reach the supposed algorithmic threshold for the planted clique problem.

A distributed algorithm for directed minimum-weight spanning tree

In the directed minimum spanning tree problem (DMST, also called minimum weight arborescence), we are given a directed weighted graph, and a root node r. Our goal is to construct a minimum-weight directed spanning tree, rooted at r and oriented outwards. We present the first sub-quadratic DMST algorithm in the distributed \({\mathsf {CONGEST}}\) network model, where the messages exchanged between the network nodes are bounded in size. We consider three versions of the model: a network where the communication links are bidirectional but can have different weights in the two directions; a network where communication is unidirectional; and the Congested Clique model, where all nodes can communicate directly with each other. Our DMST algorithm is based on a variant of Lovász’ DMST algorithm for the PRAM model, and uses a distributed single-source shortest-path (SSSP) algorithm for directed graphs as a black box. In the bidirectional \({\mathsf {CONGEST}}\) model, our algorithm has roughly the same running time as the SSSP algorithm that is used as a black box; using the state-of-the-art SSSP algorithm due to Chechik and Mukhtar (in: Symposium on principles of distributed computing (PODC), ACM, 2020, pp 464–473), we obtain a running time of \({\widetilde{O}}(\sqrt{n}D^{1/4}+D))\) rounds for the bidirectional communication case. For the unidirectional communication model we give an \({\widetilde{O}}(n)\) algorithm, and show that it is nearly optimal. And finally, for the Congested Clique, our algorithm again matches the best known SSSP algorithm: it runs in \({\widetilde{O}}(n^{1/3})\) rounds. On the negative side, we adapt an observation of Chechik in the sequential setting to show that in all three models, the DMST problem is at least as hard as the (s, t)-shortest path problem. Thus, in terms of round complexity, distributed DMST lies between single-source shortest path and (s, t)-shortest path.

An effective branch-and-bound algorithm for the maximum [formula omitted]-bundle problem

An s-bundle (where s is a positive integer) is a connected graph, the vertex connectivity of which is at least n−s, where n is the number of vertices in the graph. As a relaxation of the classical clique model, the s-bundle is relevant for representing cohesive groups with an emphasis on the connectivity of members; thus, it is of great practical importance. In this work, we investigate the fundamental problem of finding the maximum s-bundle from a given graph and present an effective branch-and-bound algorithm for solving this NP-hard problem. The proposed algorithm is distinguished owing to its new multi-branching rules, graph coloring-based bounding technique, and reduction rules using structural information. The experiments indicate that the algorithm outperforms the best-known approaches on a wide range of well-known benchmark graphs for different s values. In particular, compared with the popular Russian Doll Search algorithm, the proposed algorithm almost doubles the success rate of solving large social networks in an hour when s=5.

Hardness and tractability of the γ-Complete Subgraph problem

Finding cohesive subgraphs is a fundamental problem for the analysis of graphs. Clique is a classical model of cohesive subgraph, but several alternative definitions have been given in the literature. In this paper we consider the γ-complete graph model, which is based on relaxing the degree constraint of the clique model, that is G[S] is a γ-complete graph, with γ∈(0,1], if and only if every vertex of S has degree at least γ(|S|−1) in G[S]. In this contribution, we investigate the complexity of the problem that asks for the γ-complete subgraph of maximum order (that is, maximum number of vertices). We show that the problem is W[1]-hard for 12≤γ<1 when the parameter is order of the subgraph. Moreover, we show that the problem is fixed-parameter tractable when parameterized by the h-index of the input graph, thus solving an open question in the literature.

DMP: Content Delivery With Dynamic Movement Pattern in Vehicular Networks

Vehicular ad-hoc networks (VANETs) have been widely studied in intelligent transportation. Content delivery is an important topic that attracts many researchers. Due to vehicles that may have intermittent connections and uncertain routes, it is difficult to select an appropriate node. In this paper, we analyze the movement pattern of vehicles from real taxis’ trajectories and propose a framework for delivery prediction, which aims to select appropriate nodes. First, we propose the framework which consists of a contact clique model, a social clique model, and a prediction model based on Markov chains, to characterize the movement pattern of vehicles. Second, we capture dynamic movement patterns by dividing the time requirement into equal length slots and construct clique sequences. Based on the fact that the sociality of nodes has strong temporal correlations, we utilize the prediction model to derive future cliques and evaluate two kinds of delivery performance in the future. Finally, we design a content delivery algorithm with dynamic movement pattern (DMP) to select the appropriate node. In our experiment, DMP performs better than that of other methods in terms of overhead, average hops. Also, as the number of nodes increases, our algorithm keeps small fluctuations in node sociality.

Simple, Deterministic, Constant-Round Coloring in Congested Clique and MPC

We settle the complexity of the $(\Delta+1)$-coloring and $(\Delta+1)$-list coloring problems in the \sf CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems runn...

The role of randomness in the broadcast congested clique model

We study the role of randomness in the broadcast congested clique model. This is a message-passing model of distributed computation where the nodes of a network know their local neighborhoods and they broadcast, in synchronous rounds, messages that are visible to every other node.This works aims to separate three different settings: deterministic protocols, randomized protocols with private coins, and randomized protocols with public coins. We obtain the following results:•If more than one round is allowed, public randomness is as powerful as private randomness.•One-round public-coin algorithms can be exponentially more powerful than deterministic algorithms running in several rounds.•One-round public-coin algorithms can be exponentially more powerful than one-round private-coin algorithms.•One-round private-coin algorithms can be exponentially more powerful than one-round deterministic algorithms.

Distributed Approximation Algorithms for Steiner Tree in the CONGESTED CLIQUE

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding processors. Regarding output, each processor should know whether the vertex assigned to it is in the solution or not and which of its incident edges are in the solution. We present two deterministic distributed approximation algorithms for the Steiner tree problem in the CCM. The first algorithm computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages for a given connected undirected weighted graph of [Formula: see text] nodes. Note here that [Formula: see text] notation hides polylogarithmic factors in [Formula: see text]. The second one computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] and [Formula: see text] are the shortest path diameter and number of edges respectively in the given input graph. Both the algorithms achieve an approximation ratio of [Formula: see text], where [Formula: see text] is the number of leaf nodes in the optimal Steiner tree. For graphs with [Formula: see text], the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with [Formula: see text], the second algorithm outperforms the first one in terms of the round complexity. In fact when [Formula: see text] then the second algorithm achieves a round complexity of [Formula: see text] and message complexity of [Formula: see text]. To the best of our knowledge, this is the first work to study the Steiner tree problem in the CCM.