Aggregability is NP-Hard

Many dynamical systems are aggregable in the sense that we can divide their variables x1,...,xn into several (k) non-intersecting groups and find combinations y1,...,yk of variables from these groups (macrovariables) whose dynamics depend only on the initial values of the macrovariables. For very large systems, finding such an aggregation is often the only way to perform a meaningful analysis of such systems. Since aggregation is important, researchers have been trying to find a general efficient algorithm for detecting aggregability. In this paper, we show that in general, detecting aggregability is NP-hard even for linear systems, and thus (unless P=NP), we can only hope to find efficient detection algorithms for specific classes of systems. We also show that in the linear case, once the groups are known, it is possible to efficiently find appropriate combinations ya

Aggregability is NP-Hard

Many dynamical systems are aggregable in the sense that we can divide their variables x1,..., xn into several (k) non-intersecting groups and find combinations y1,..., yk of variables from these groups (macrovariables) whose dynamics depend only on the initial values of the macrovariables. For very large systems, finding such an aggregation is often the only way to perform a meaningful analysis of such systems. Since aggregation is important, researchers have been trying to find a general efficient algorithm for detecting aggregability. In this paper, we show that in general, detecting aggregability is NP-hard even for linear systems, and thus (unless P=NP), we can only hope to find efficient detection algorithms for specific classes of systems. We also show that in the linear case, once the groups are known, it is possible to efficiently find appropriate combinations ya. What is aggregability. Many systems in nature can be described as dynamical systems, in which the state of a system at each moment of time is characterized by the values of (finitely many) variables x1,..., xn, and the change of the state over time is described by an equation x ′ i = fi(x1,..., xn), where • for continuous-time systems, in which the time t can take any real value, x ′ i is the first time derivative of xi: dxi dt = fi(x1,..., xn); (1a) • for discrete-time systems, in which the time t can only take integer values, x ′ i is the value of xi at the next moment of time: ∗ c○V. Kreinovich and M. Shpak, 2006 xi(t + 1) = fi(x1(t),..., xn(t)). (1b

Decomposable Aggregability in Population Genetics and Evolutionary Computations: Algorithms and Computational Complexity

Another example of a system where detecting aggregability is important is a one that describes the dynamics of an evolutionary algorithm- which is formally equivalent to models from population genetics. For very large systems, finding such an aggregation is often the only way to perform a meaningful analysis of such systems. Since aggregation is important, researchers have been trying to find a general efficient algorithm for detecting aggregability. In this chapter, we show that in general, detecting aggregability is NP-hard even for linear systems, and thus (unless P=NP), we can only hope to find efficient detection algorithms for specific classes of systems. Moreover, even detecting approximate aggregability is NP-hard. We also show that in the linear case, once the blocks are known, it is possible to efficiently find appropriate linear combinations ya. 1 What is Aggregability Many systems in nature can be described as dynamical systems, in which thestate of a system at each moment of time is characterized by the values of (finitely many) variables x1,..., xn, and the change of the state over time isdescribed by an equation x0i = fi(x1,..., xn), where * for continuous-time systems, in which the time t can take any real value, x0i is the first time derivative of xi