In this paper, we show that the Max Cut problem is NP-complete on interval graphs.

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The ARBITRARY PATTERN FORMATION problem asks to design a distributed algorithm that allows a set of autonomous mobile robots to form any specific but arbitrary geometric pattern given as input. The problem has been extensively studied in literature in continuous domains. This paper investigates a discrete version of the problem where the robots are operating on a two dimensional infinite grid. The robots are assumed to be autonomous, identical, anonymous and oblivious. They operate in Look-Compute-Move cycles under a fully asynchronous scheduler. The robots do not agree on any common global coordinate system or chirality. We have shown that a set of robots can form any arbitrary pattern, if their starting configuration is asymmetric.

The Arbitrary Pattern Formation problem asks for a distributed algorithm that moves a set of autonomous mobile robots to form any arbitrary pattern given as input. The robots are assumed to be autonomous, anonymous and identical. They operate in Look-Compute-Move cycles under a fully asynchronous scheduler. The robots do not have access to any global coordinate system. The existing literature that investigates this problem, considers robots with unobstructed visibility. This work considers the problem in the more realistic obstructed visibility model, where the view of a robot can be obstructed by the presence of other robots. The robots are assumed to be punctiform and equipped with visible lights that can assume a constant number of predefined colors. We have studied the problem in two settings based on the level of consistency among the local coordinate systems of the robots: two axis agreement (they agree on the direction and orientation of both coordinate axes) and one axis agreement (they agree on the direction and orientation of only one coordinate axis). In both settings, we have provided a full characterization of initial configurations from where any arbitrary pattern can be formed.

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