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We give logarithmic lower bounds for the approximability of the Minimum Dominating Set problem in connected (α, β)-Power Law Graphs. We give also a best up to now upper approximation bound on the problem for the case of the parameters β > 2. We develop also a new functional method for proving lower approximation bounds and display a sharp phase transition between approximability and inapproximability of the underlying problem. This method could also be of independent interest.

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In this paper we study the special case of Graphic TSP where the underlying graph is a power law graph (PLG). We give a refined analysis of some of the current best approximation algorithms and show that an improved approximation ratio can be achieved for certain ranges of the power law exponent β. For the value of power law exponent β = 1.5 we obtain an approximation ratio of 1.34 for Graphic TSP. Moreover we study the (1, 2)-TSP with the underlying graph of 1-edges being a PLG. We show improved approximation ratios in the case of underlying deterministic PLGs for β greater than 1.666. For underlying random PLGs we further improve the analysis and show even better expected approximation ratio for the range of β between 1 and 3.5. On the other hand we prove the first explicit inapproximability bounds for (1, 2)-TSP for an underlying power law graph., where E k is the expected lower bound on the number of 2-edges in an optimum tour.On the other hand we show that for β > 1, the Power Law (1, 2)-TSP is NP-hard to approximate within approximation ratio (ζ(β)+ 1 /2)·3 β−1 ·2·(β−1)·354+1 (ζ(β)+ 1 /2)·3 β−1 ·2·(β−1)·354 . This gives an approximation lower bound of 1.00086 for β = 1.2 and of 1.0012 for β = 1.1. Organization of the PaperIn Section 2 we give an outline of the methods and constructions used in the paper. Section 3 provides the definition of the PLG model due to [ACL01] and related notations. In Section 4 we describe our results for the Graphic TSP on power law graphs. In Section 5 we present our results on the (1, 2)-TSP with underlying power law graph. Section 5.1 contains the analysis of the algorithm from [PY93] for deterministic power law graphs. Section 5.2 deals with the case of a large power law exponent. In Section 5.3 we consider the case when the underlying graph is a random PLG. Approximation lower bounds are given in Section 5.4. 2 8

In this paper we study the MAX-CUT problem on power law graphs (PLGs) with power law exponent β. We prove some new approximability results on that problem. In particular we show that there exist polynomial time approximation schemes (PTAS) for MAX-CUT on PLGs for the power law exponent β in the interval (0, 2). For β > 2 we show that for some ε > 0, MAX-CUT is NP-hard to approximate within approximation ratio 1 + ε, ruling out the existence of a PTAS in this case. Moreover we give an approximation algorithm with improved constant approximation ratio for the case of β > 2.