We prove that the problem of computing an Arrow-Debreu market equilibrium is PPAD-complete even when all traders use additively separable, piecewise-linear and concave utility functions. In fact, our proof shows that this market-equilibrium problem does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time.

We study the problem of solving simple stochastic games, and give both an interesting new algorithm and a hardness result. We show a reduction from fine approximation of simple stochastic games to coarse approximation of a polynomial sized game, which can be viewed as an evidence showing the hardness to approximate the value of simple stochastic games. We also present a randomized algorithm that runs inÕ( √ |V R |!) time, where |V R | is the number of RANDOM vertices andÕ ignores polynomial terms. This algorithm is the fastest known algorithm when |V R | = ω(log n) and |V R | = o( √ min |V min |, |V max |) and it works for general (non-stopping) simple stochastic games.

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a b s t r a c tWe study the minimum weight dominating set problem in weighted unit disk graph, and give a polynomial time algorithm with approximation ratio 5 + , improving the previous best result of 6 + in [Yaochun Huang, Xiaofeng Gao, Zhao Zhang, Weili Wu, A better constant-factor approximation for weighted dominating set in unit disk graph, J. Comb. Optim. (ISSN: 1382-6905) (2008) 1573-2886. (Print) (Online)]. Combining the common technique used in the above mentioned reference, we can compute a minimum weight connected dominating set with approximation ratio 9 + , beating the previous best result of 10 + in the same work.