arXiv subjects

Susumu Kubo

Publications and source records attributed to Susumu Kubo.

4 recordsLinked to original sources

arXiv2026

The basic tropical polynomials generate the semifield of $r$-symmetric tropical rational functions

Let the symmetric group $S_n$ act on the space of $n \times r$ real matrices by permuting rows, so orbits are multisets of $n$ points in $\mathbb{R}^r$. The basic $r$-symmetric tropical polynomials form a family of $\binom{n+r}{r}-1$ nonconstant invariants of degree at most $n$ that separates orbits and embeds the orbit space bi-Lipschitzly. We prove that this family generates the semifield of all $r$-symmetric tropical rational functions, answering a question raised in [J. Pure Appl. Algebra 223 (2019) 72-85]. Derksen showed that the invariant semifield of any permutation group $G \le S_N$ is generated in degree at most $N p_1 \cdots p_{|G|}$ ($p_i$ the $i$th prime), which for the row action is $nr p_1 \cdots p_{n!}$; the present result replaces this by generators of degree at most $n$. The generating expression is a finite minimum over the ways of re-assembling a multiset from its sorted columns, with penalties from the basic values that, via the bi-Lipschitz inequality, dominate a wrong re-assembly. The same penalties describe the image of the basic coordinate map as the zero set of a single tropical rational function and yield an expression algorithm. Subfamilies of the basic family containing the single-column values generate if and only if they separate. For any permutation group $G \le S_N$ the same mechanism generates the invariant semifield in degree at most $\max\{N, \binom{N}{2}\}$, a quadratic bound independent of the group order; combined with a genericity theorem of Cahill, Iverson, Mixon, and Packer, it yields $2N+1$ invariant tropical polynomials that separate orbits and $3N$ that generate, with at least $N$ necessary for each task. The quadratic bound is optimal: every $A_N$-invariant tropical polynomial of degree less than $\binom{N}{2}$ is $S_N$-invariant, so every separating family for the alternating group $A_N$ contains a member of degree at least $\binom{N}{2}$.

math.AG↗

arXiv2026

Partially Ordered Sets Corresponding to the Partition Problem

The partition problem is a well-known NP-complete problem. We focus on its optimization version. We propose two partially ordered sets (posets) corresponding to the partition problem and develop an order-theoretic framework for solving it. The first poset is order-isomorphic to a well-known poset whose structure is related to solutions of the subset sum problem, while the second is a subposet of the first and plays a crucial role in this paper. The partial order characterizes the dominance relations between subsets that hold uniformly across all instances. We first show several properties of the two posets, such as size, height, and width (the size of the largest antichain, i.e., the largest set of pairwise incomparable elements). The two posets have the same width, which is $Θ(2^n / n^{3/2})$ for $n$ congruent to $0$ or $3$ modulo $4$; this exponential width indicates the hardness of the partition problem. We then prove that the initial candidate solutions are the elements of the second poset, whose size is $2^{n} - 2 \binom{n}{\lfloor n/2 \rfloor}$. Since a partition corresponds to two elements of the poset, the number of initial candidate partitions is half of that, i.e., $2^{n-1} - \binom{n}{\lfloor n/2 \rfloor}$. We prove that the candidate solutions can be further reduced based on the partial order, and we establish a necessary and sufficient condition, phrased in terms of the second poset, for a subset to attain the optimal value. Building on this optimality criterion, we finally derive several polynomially solvable cases from the structure of the second poset. %considering the minimal and maximal elements of the second poset. Our approach offers a useful tool for structural analysis of the partition problem.

cs.DM↗

arXiv2026

Stable complete coordinates for multisets of points via basic $r$-symmetric tropical polynomials

A multiset of $n$ unordered points in $\mathbb{R}^r$ -- a point cloud, or, for $r=2$, a persistence barcode of birth-death pairs -- is a point of the orbit space $\mathbb{R}^{nr}/S_n$ for the symmetric group $S_n$ permuting the rows of an $n \times r$ matrix; a separating family of invariants on this space is exactly a complete set of permutation-independent coordinates. We provide one that is explicit, small, and stable, in the max-plus (tropical) setting: for all $n \geq 1$ and $r \geq 1$, the $\binom{n+r}{r}$ basic $r$-symmetric tropical polynomials, of degree at most $n$, separate the orbits of $S_n$ on $\mathbb{R}^{nr}$. This settles in full a problem left open in [Kubo, J. Pure Appl. Algebra 223 (2019) 72-85], where separation was known only for $r=2$ and special cases of $r \geq 3$, and yields a family far smaller and of lower degree than the general separating sets from Derksen's recent theory of tropical invariants for permutation actions ($nr + (nr)!/n!$ invariants of degree $O(n^2 r^2)$). The proof is elementary and constructive: the basic values are identified with a transportation problem, and the multiset is recovered from the dual by an explicit algorithm. We further show the coordinate map is a bi-Lipschitz embedding for all $n$ and $r$, being an injective max filter bank (via the bi-Lipschitz theory of max filtering), with an explicit Lipschitz constant for the forward bound and a fully explicit, dimension-free distortion when $r=1$. Finally we determine when the pairwise values suffice (exactly $n \leq 3$) and show that invariants on at least three columns and of degree less than $n$ are necessary in general, the obstruction being a standard non-uniqueness configuration from discrete tomography.

cs.DM↗

arXiv2024

Composition Orderings for Linear Functions and Matrix Multiplication Orderings

We consider composition orderings for linear functions of one variable. Given $n$ linear functions $f_1,\dots,f_n$ and a constant $c$, the objective is to find a permutation $σ$ that minimizes/maximizes $f_{σ(n)}\circ\dots\circ f_{σ(1)}(c)$. It was first studied in the area of time-dependent scheduling, and known to be solvable in $O(n\log n)$ time if all functions are nondecreasing. In this paper, we present a complete characterization of optimal composition orderings for this case, by regarding linear functions as two-dimensional vectors. We also show several interesting properties on optimal composition orderings such as the equivalence between local and global optimality. Furthermore, by using the characterization above, we provide a fixed-parameter tractable (FPT) algorithm for the composition ordering problem for general linear functions, with respect to the number of decreasing linear functions. We next deal with matrix multiplication orderings as a generalization of composition of linear functions. Given $n$ matrices $M_1,\dots,M_n\in\mathbb{R}^{m\times m}$ and two vectors $w,y\in\mathbb{R}^m$, where $m$ denotes a positive integer, the objective is to find a permutation $σ$ that minimizes/maximizes $w^\top M_{σ(n)}\dots M_{σ(1)} y$. The problem is also viewed as a generalization of flow shop scheduling through a limit. By this extension, we show that the multiplication ordering problem for $2\times 2$ matrices is solvable in $O(n\log n)$ time if all the matrices are simultaneously triangularizable and have nonnegative determinants, and FPT with respect to the number of matrices with negative determinants, if all the matrices are simultaneously triangularizable. As the negative side, we finally prove that three possible natural generalizations are NP-hard: 1) when $m=2$, 2) when $m\geq 3$, and 3) the target version of the problem.

cs.DS↗