ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2013, Vol. 7, No. 2, pp. 142–152. c  Pleiades Publishing, Ltd., 2013. Original Russian Text c  A.I. Erzin, P.V. Plotnikov, Yu.V. Shamardin, 2013, published in Diskretnyi Analiz i Issledovanie Operatsii, 2013, Vol. 20, No. 1, pp. 12–27. On Some Polynomially Solvable Cases and Approximate Algorithms in the Optimal Communication Tree Construction Problem A. I. Erzin 1,2* , R. V. Plotnikov 2** , and Yu. V. Shamardin 1*** 1 Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia 2 Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia Received January 17, 2012; in final form, March 28, 2012 Abstract—Considering an arbitrary undirected n-vertex graph with nonnegative edge weights, we seek to construct a spanning tree minimizing the sum over all vertices of the maximal weights of the incident edges. We find some particular cases of polynomial solvability and show that the minimal span whose edge weights lie in the closed interval [a, b] is a ( 2 − 2a a+b+2b/(n−2) ) -approximate solution, and the problem of constructing a 1.00048-approximate solution is NP-hard. We propose a heuristic polynomial algorithm and perform its a posteriori analysis. DOI: 10.1134/S1990478913020038 Keywords: wireless communication network, spanning tree, approximate algorithm INTRODUCTION Many communication networks use wireless communication for information exchange. The energy losses at their elements are proportional to d s with s ≥ 2, where d is the transmission distance [3]. In some networks, for instance in wireless sensor networks, the elements (sensors) have restricted energy reserves, and efficient energy use by them makes it possible to extend the network’s lifetime [1, 10, 13, 14]. To save on energy use, the modern sensors are capable of adjusting the radio transmission distance. This leads to the problem of determining the transmission distance for every element of the network so that to minimize the total energy required to keep the graph connected. If we assume that the radio signal propagates in the same way in all directions then all elements in the transmission zone (not beyond the transmission distance) receive the message. In this case, we may assume that the communication network, a spanning subgraph whose edges carry the transmission, is a complete graph [3, 6, 9, 10]. However, it is not always true that the signal propagates in the same way in all directions and at all distances. Therefore, in general, we should assume that the communication graph G =(V,E) can be an arbitrary spanning subgraph and the energy losses for the transmission along the edges may vary. Thus, if c ij ≥ 0 stands for the energy loss incurred in data transmission from i ∈ V to j ∈ V then, in the connected subgraph T =(V,E  ) with E  ⊆ E, the energy loss at the vertex i ∈ V equals E i (T )= max j |(i,j )∈E  c ij . The goal of this article is to study the problem of constructing a spanning subgraph T for which the sum ∑ i∈V E i (T ) is minimal. Without loss of generality, we may assume that T is a spanning tree. As we noted above, the problems of this kind arise, for instance, in wireless sensor networks, when the location of sensors is known and we are required to determine an energy-efficient graph connecting all sensors [13]. It is customary in the literature to consider as the communication graph of a sensor * E-mail: adilerzin@math.nsc.ru ** E-mail: nomad87@ngs.ru *** E-mail: orlab@math.nsc.ru 142