Computational complexity of the ground-state determination of atomic clusters

Genetic Algorithms in Application to the Geometry Optimization of Nanoparticles

“…Even though this important problem of chemical physics does not resemble TSP at a first glance, a deeper analysis shows that these two problems are indeed very similar combinatorial problems if the solution space of the nanoparticle problem is constructed as being composed of stable isomers only. The number of stable isomers of the nanoparticle increases exponentially just like the possible paths in TSP as the system size increases and an exact solution becomes intractable even for moderate sized nanoparticles [7].…”

NP-hardness of the cluster minimization problem revisited

“…[9], a celebrated work that is frequently cited in the literature as providing a proof that the "cluster minimization problem" is NP-hard (see e.g. [10,11,12,13,14,15]).…”

“…by showing that every instance I ′ of π ′ can be transformed to an instance I = f (I ′ ) of π in polynomial time, and that a solution of I also solves I ′ . The NPhardness proof of Wille and Vennik was based on such a transformation [9]: Formulating the cluster minimization problem in terms of "graphs" (see e.g. [4]), these authors considered what will be henceforth referred to as the WEIGHTED EDGE problem, showing that the TRAVELING SALESPERSON (TSP) problem can be transformed to WEIGHTED EDGE; since TSP is known to be NP-hard, so must be WEIGHTED EDGE.…”

“…When formulating the cluster minimization problem as WEIGHTED EDGE, the authors of Ref. [9] had in mind the following analogy: V is the set of "sites" that the N particles are allowed to occupy, and for every pair of sites with edge e = (v i , v j ), the function w(e) = w(v i , v j ) measures the strength of the interaction between two hypothetical particles placed in these sites. The set V is of course discrete, reflecting the discretization of the problem that accompanies any computational method, but notice that no specific structure is given to either V or w(e) (this arbitrariness will be discussed later in the text).…”

“…[9] by showing that it "contains" TSP, more precisely by showing that there exists a weight function w(e), a choice of |V | as a function of N, and a labeling of the vertices in terms of pairs of "cities," such that a solution of WEIGHTED EDGE would yield a minimum TSP tour. ‡ Though not stated in [9], it is easy to check that this labeling and ‡ A more direct proof that WEIGHTED EDGE is NP-hard can be given. Indeed, given any graph choice of w(e) can be done in polynomial time given any instance of TSP.…”