arXiv subjects

Yoshiteru Ishida

Publications and source records attributed to Yoshiteru Ishida.

4 recordsLinked to original sources

arXiv2026

A Quadratic Lower Bound for the Shield Number of the Stable Marriage Problem: Rearrangement, Extremal Construction, and Biclique Realizability

We study the Shield number of the stable marriage problem, defined as the minimum, over all preference profiles of size n, of the maximum number of blocking pairs attainable by a complete matching. We establish the universal quadratic lower bound $σ(n) \ge \lceil n(n-2)/6 \rceil$ using a rearrangement inequality and show that this bound is the strongest consequence obtainable from uniform first-moment arguments. We further prove a sharp min-rearrangement inequality and construct an explicit cyclic Hollow-Shell instance attaining $\lfloor (n-1)^2/4 \rfloor$ blocking pairs. We also establish sufficient Hall-type conditions for biclique realizability, prove structural limitations of uniform averaging and two-parameter linear assignment objectives, and provide complete proofs for all unconditional results. Computer-assisted verification is used only for explicitly identified computational statements and is clearly distinguished from mathematical proofs.

math.CO↗

arXiv2026

A Reachable-State Operator Formulation of Deferred Acceptance: Progress Invariants and Structural Diagnostics

We give a reachable-state operator formulation of one-to-one deferred acceptance with strict, possibly incomplete preference lists. A local update operator is defined for each active proposer, while a scheduler selects which local update is applied. On the states reachable from the canonical empty initial state, three invariants are immediate: the set of proposed edges grows strictly, every proposer visits each acceptable receiver at most once, and every receiver holds its most-preferred proposal received so far. These invariants yield termination after at most |E| proposals and stability of every terminal reachable state. The classical rejection lemma then gives proposer optimality and schedule independence. We separate these trajectory statements from two logically different results: the distributive-lattice structure of the full stable-matching set and one-sided strategy-proofness. A diagnostic table records which proof obligation is lost when a model changes the bipartition, ordinal comparisons, proposal irreversibility, or receiver choice rule. The paper makes no new complexity claim; its purpose is a precise operator-level account of the classical proof architecture and of the limits of that architecture.

cs.DS↗

arXiv2026

Regular anti-phase templates in the stable marriage problem: a generator criterion, its converse, and a counting bound

We study a family of highly symmetric instances of the stable marriage problem built from regular actions of finite groups. Given a finite group G of order n and an ordering A of its elements, we define the regular anti-phase template P(G,A). These templates have n canonical stable matchings. We show that the anti-phase condition is canonical: among automorphism-maximal profiles, the anti-phase templates are exactly those satisfying a constant rank-sum identity. This gives a structural characterization rather than an ad hoc definition. We prove a generator criterion and its exact converse: the stable set has size n if and only if each adjacent quotient generates the group. This result holds for all finite groups and does not require commutativity. We further establish a counting lower bound for the number of stable matchings in terms of subgroup indices. The bound is sharp for groups of order at most 5 and for all groups of order 4; in particular, it yields at least 10 stable matchings for the Klein group, with equality confirmed by enumeration. Finally, we show that cyclic profiles do not always produce chains; the structure depends on the ordering. All computational claims are verified by an accompanying script.

math.CO↗

arXiv2026

Euler Constraints and the Cubic Criticality of Complete Bipartite Preference Structures

We introduce a polyhedral realizability problem for complete bipartite preference structures, the two-sided strict preference profiles that underlie the stable marriage problem. Given two parts M and W of size k, a strong polyhedral realization asks for a bipartite polyhedral graph with bipartition classes of size k in which every vertex is adjacent to all but one vertex on the opposite side. This condition attempts to represent complete opposite-side preference capacity by direct polyhedral adjacency, with one geometrically exceptional opposite-side vertex for each agent. We prove an Euler-type criticality theorem: such a strong polyhedral realization exists if and only if k=4, and in that case the underlying graph is the cube graph Q_3, equivalently K_{4,4} minus a perfect matching. The proof is the collision of the strong edge requirement E=k(k-1) with the bipartite planar bound E<=4k-4 and the minimum-degree constraint for polyhedral graphs. We then distinguish weak realizations. A family of (k-1)-gonal trapezohedra gives balanced bipartite polyhedral graphs with 2k vertices and 4k-4 edges for all k>=4, showing that weak maximal realizability persists beyond the cubic critical case. However, the Euler interval 3k<=E<=4k-4 is not fully realizable: we give a direct proof that no bipartite polyhedral graph exists with V=10 and E=15, equivalently that there is no even triangulation of the sphere on seven vertices. Finally, in the cube case, we show that cube-distance compatibility of a size-four preference profile is equivalent to the existence of a perfect matching of mutually last-ranked pairs. This gives a first preference-theoretic manifestation of the cubic criticality.

math.CO↗