The additively separable hedonic game (ASHG) is a model of coalition formation games on graphs. In this paper, we intensively and extensively investigate the computational complexity of finding several desirable solutions, such as a Nash stable solution, a maximum utilitarian solution, and a maximum egalitarian solution in ASHGs on sparse graphs including boundeddegree graphs, bounded-treewidth graphs, and near-planar graphs. For example, we show that finding a maximum egalitarian solution is weakly NP-hard even on graphs of treewidth 2, whereas it can be solvable in polynomial time on trees. Moreover, we give a pseudo fixed parameter algorithm when parameterized by treewidth.

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In this paper, a 2-D binary-valued (BV) lapped transform (LT) is proposed. The proposed LT has basis images which take only BV elements and satisfies the axial-symmetric (AS) property. In lD, there is no 2-point LT with the symmetric basis vectors. Hence, in 2D, there is no 2 x 2-point separable ASLT. By taking non-separable BV basis images, this paper shows that a 2 x 2-point ASLT can be obtained. For its similarity to the Hadamard transform (HT), this paper refer to it as the lapped Hadamard transform (LHT). The efficient implementation is also given.

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