Information Processing Letters 46 (1993) 169-172 Elsevier 35 June 1993 Approximating the minimum maximal independence number Magnh M. Halldhsson zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA School zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of Information Science, Japan Adcanced Institute of Science and Technology . Tatsunokuchi, Ishikawn 923- 12, Japan Communicated by M.J. Atallah Received 26 January 1993 Revised 15 April 1993 Abstract Halld6rsson. M.M., Approximating the minimal maximal independence number. Information Processing Letters 16 (1993) 169-172. We consider the problem of approximating the size of a minimum non-extendible independent set of a graph. also known as the minimum dominating independence number. We strengthen a result of Irving to show that there is no constant E > 0 for which this problem can be approximated within a factor of n’-’ m polynomial time, unless P = NP. This is the strongest lower bound we are aware of for polynomial-time approximation of an unweighted NP-complete graph problem. Keywords: Combinatorial problems; approximation algorithms 1. Introduction A maximal independent set in an undirected graph is a set of mutually non-adjacent vertices such that the introduction of an additional vertex destroys the non-adjacency property. The mini- mum maximal independence number (MMIN) of a graph is the size of the smallest such set. It is also commonly known as the size of a minimum inde- pendent dominating set. Determining this num- ber for arbitrary graphs is known to be NP-com- plete. In this paper, we show that assuming P f NP, MMIN cannot be approximated in polynomial time within a factor of n’-’ for any E > 0, where Correspondence to: M. Halldbrsson, School of Information Science, Japan Advanced Institute of Science and Technol- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE ogy, Tatsunokuchi, Ishikawa 923-12, Japan. Email: magnus@ jaist-east.ac.jp. n is the number of vertices. In fact, we obtain a spectrum of results, with tradeoffs between as- sumptions on the solvability of the satisfiability problem and lower bounds on the approximation of MMIN. This work is a follow-up to a paper of Irving [2], who showed that MMIN could not be approx- imated within any constant factor (assuming P # NP). Our basic result is a simple generalization of his construction. On a related note, Kann [3] has since shown that the MMIN problem is complete for the class of polynomial-bounded minimization problems, implying, among other things, a some- what weaker lower bound on the approximability of MMIN of n’, for some constant S > 0. 1.1. Notation For an algorithm (or a function) A, let A(G) denote the number output on input G, represent- OOZO-0190/93/SO6.00 0 1993 - Eisevier Science Publishers B.V. All rights reserved 169