Andrei Popa | Search arXiv

arXiv2026

Strong NP-Completeness of Unrestricted Balanced Mobiles

A mobile is a rooted full binary tree whose leaves carry positive integer weights. The imbalance of an internal node is the absolute difference between the total weights of its two child subtrees, and the cost of the mobile is the sum of these imbalances. In the unrestricted \emph{Balanced Mobiles} problem, only the multiset of leaf weights is given: both the tree topology and the placement of the weights must be chosen so as to minimize the cost. The computational complexity of this unrestricted variant has remained open, although the variant with a prescribed topology is strongly NP-hard. We close this gap by proving that the decision version of unrestricted Balanced Mobiles is strongly NP-complete. Our reduction from Numerical 3-Dimensional Matching with Distinct Integers uses three widely separated numerical scales. Tight telescoping bounds force every threshold-achieving mobile into a canonical hierarchy, after which pairwise distinctness of the source integers collapses the hierarchy to single triples from which a valid numerical matching can be recovered.

cs.CC↗

arXiv2026

Complexity and Algorithms for Unary Translocation Distance

Given a finite set of integers $A$, a \emph{unary translocation} produces a new set $A' = A \cup \{u,v\}$, where $u$ and $v$ are nonnegative integers satisfying $x+y=u+v$ for some $x,y\in A$. For an input set $A$ and a target set $B$, the \emph{unary translocation distance} is the minimum number of unary translocations required to obtain a superset containing $B$. In this paper, we study this problem from both theoretical and computational perspectives. We prove that computing the unary translocation distance is strongly NP-hard, thereby answering an open question raised by \citet{ConstantinMiclausPopa2026UnaryTranslocation}. On the positive side, we give an exact pseudo-polynomial algorithm for every fixed constant value of $|B|$, extending our previous results for $|B|\leq 2$. For arbitrary target sets, we present a $2$-approximation algorithm, an additive $(|B|-1)$-approximation algorithm, and show that the additive algorithm also yields a $3$-approximation. We also propose parameterized algorithms, including algorithms parameterized by the maximum value in the input set together with the optimum distance, and by the maximum value in the target set together with $|B|$. In addition, we propose an integer linear programming formulation that gives an exact mathematical model for the problem, analyze its size, and show that the LP relaxation has integrality gap at least $\frac{4}{3}$. Finally, we report computational experiments comparing the $2$-approximation algorithm, beam search, and simulated annealing. The results show that the approximation algorithm is highly effective in practice and often outperforms the heuristic baselines.

cs.DS↗