Mathematical Problems in Engineering

Volume 2016, Article ID 9790629, 10 pages

http://dx.doi.org/10.1155/2016/9790629

## A New Distributed Approximation Algorithm for the Maximum Weight Independent Set Problem

^{1}College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210023, China^{2}National Mobile Communications Research Laboratory, Southeast University, Nanjing, China

Received 9 March 2016; Revised 29 July 2016; Accepted 9 August 2016

Academic Editor: Dan Simon

Copyright © 2016 Peng Du and Yuan Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Maximum weight independent set (MWIS) is a combinatorial optimization problem that naturally arises in many applications especially wireless networking. This paper studies distributed approximation algorithms for finding MWIS in a general graph. In the proposed algorithm, each node keeps exchanging messages with neighbors in which each message contains partial solutions of the MWIS optimization program. A parameter is introduced to achieve different tradeoff between approximation accuracy and space complexity. Theoretical analysis shows that the proposed algorithm is guaranteed to converge to an approximate solution after finite iterations; specifically, the proposed algorithm is guaranteed to converge to the optimal solution with . Simulation results confirm the effectiveness of the proposed distributed algorithm in terms of weight sum, message size, and convergence performance.

#### 1. Introduction

Consider a graph with a set of nodes and a set of edges. For each node , there is a positive weight . A subset of can be represented by binary variable , , where is 1 if is in the subset and 0 otherwise. A subset is called an independent set if no two nodes in the subset are connected by an edge. We are interested in finding the maximum weight independent set (MWIS) [1], which can be expressed as an integer program:

The MWIS problem has been extensively studied in the literature. For example, it is known to be solvable in polynomial time for many cases including perfect graphs [2], interval graphs [2], disk graphs [3], claw-free graphs [4], fork-free graphs [5], trees [6], sparse random graphs [7, 8], circle graphs [9], and growth-bounded graphs [10]. Moreover, there has been extensive work on approximating the MWIS [11], and specialized algorithms have been developed for exactly computing the MWIS [12–15].

Further, the MWIS problem naturally arises in many applications, especially wireless networking, which require distributed solutions. For example [16–18], in scenarios involving resource scheduling in wireless networks that lack a centralized infrastructure and where nodes can only communicate with local neighbors, the MWIS problem needs to be solved in a distributed manner. Fundamentally, any two wireless nodes that transmit at the same resource will interfere with each other if they are located close-by. The scheduling problem is to decide which nodes should transmit at the given resource so that there is no interference and nodes with long queue length are given priority. If each node is given a weight equal to the queue length, it is optimal to schedule the set of nodes with the highest total weight. If a conflict graph is made, with an edge between each pair of interfering nodes, the scheduling problem is exactly the MWIS problem for the conflict graph. The lack of an infrastructure and the local nature of communication require a distributed algorithm for solving the MWIS problem.

There are three types of distributed algorithms for MWIS problem: greedy algorithms, carrier sense random multiple access (CSMA) algorithms, and message-passing algorithms.

For greedy algorithms, several simple distributed algorithms for MWIS have been proposed in the literature [19–22]. These algorithms are based on the greedy principle and require only knowledge of the local topology at each node. However, due to the inherent difficulty of the MWIS problem, greedy algorithms are not guaranteed to obtain the optimal solution for general graph.

For random multiple access algorithms, several CSMA-based algorithms have been proposed [23–26]. These algorithms are distributed, are of low complexity, and are easy to implement. They use a random access scheme according to which a link seizes the channel with a probability that increases exponentially with a certain link-dependent weight. As explained in [27], these schemes perform sampling over independent sets of the graph and converge to the optimal MWIS in graphs with large weights. However, according to [28], message-passing algorithms can achieve significantly smaller aggregate long-run average queue length than CSMA algorithms.

Several message-passing algorithms for producing a feasible solution to the MWIS were proposed in [28–34]. For example, in [30], Sanghavi et al. show that a simple modification of max product becomes gradient descent on the dual of the linear programming associated with the MWIS problem and converges to the dual optimum. They also develop a message-passing algorithm that recovers the primal MWIS solution from the output of the descent algorithm and show that the MWIS estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the MWIS is unique. In [31], it has been shown that polynomial time methods exist for perfect graphs which include bipartite graphs and many others. Further, in [29], message-passing methods were developed for perfect graphs. Additionally, it is worth mentioning that message-passing methods have also been successfully applied to solve the maximum weight matching (MWM) problem [35–38].

This paper studies this problem and proposes a new distributed algorithm for the general graph. The proposed algorithm runs iteratively in which each node receives messages from its neighbors, updates the message of its own, and then broadcasts the message to the neighbors. During the message update procedure, each node shall first combine all received messages into a single one and then delete some elements from the message to control the size of the message. Theoretical analysis and computer simulation show that the proposed algorithm will converge after finite iterations and can achieve tradeoff between approximation accuracy and space complexity.

The rest of the paper is outlined as follows. In Section 2, some useful notations are introduced. In Section 3, a new distributed algorithm for MWIS is proposed. In Section 4, the theoretical analysis of the proposed distributed algorithm including convergence and optimality is presented. Section 5 reports simulation results of the proposed algorithm and its comparison to the existing distributed MWIS algorithms. Finally, concluding remarks are given in Section 6.

#### 2. Notation

This section introduces some notations which will be used throughout this paper. The symbols used in this paper are summarized in the List of Symbols.

##### 2.1. Partial Solution

Let denote the full variable set of the MWIS problem. Assuming is a subset of , a partial solution of the MWIS problem is expressed aswhere is 0 or 1. We call the partial variable set of the MWIS problem. The objective associated with is expressed asFurther, given a partial variable set , the partial solution set can be expressed aswhere

To help understand this, we use the graph shown in Figure 1 as an example. For this graph, there are four nodes and the weights are , , , and . The full variable set . Assuming the partial variable set , then is a partial solution of the MWIS problem. The objective associated with this partial solution is .