Block–HLP Prefix-Sum Capacity Constraints

• The paper introduces a framework that precisely characterizes feasible aggregate vectors using block-wise prefix-sum inequalities combined with arithmetic progression congruence conditions.
• It utilizes a sorting-and-prefix-sum methodology to efficiently verify constraint feasibility in both coalition voting scenarios and block-probe data structure models.
• The constraints reveal significant trade-offs in multiwinner elections and non-adaptive query systems by establishing tight bounds on coalition power and memory capacity redundancy.

Block--HLP Prefix-Sum Capacity Constraints constitute a precise mathematical characterization of feasible aggregate resource assignments in settings where agents or coalitions can only exert limited coordinated influence subject to ballot or data block structure. Originally formulated in the geometry of coalition power in multiwinner elections under positional scoring rules, these constraints are a system of prefix-sum inequalities based on row or block majorization, further enhanced by arithmetic progression (AP) congruence conditions in the discrete integer domain. They also emerge in the analysis of data structures for block-based prefix sums within the cell-probe model, revealing a deep connection between combinatorial ballot manipulation and memory capacity bounds for non-adaptive query systems (Guo et al., 23 Jan 2026, 0906.1370).

1. Definition and Mathematical Formulation

The Block--HLP (Hardy–Littlewood–Pólya) prefix-sum capacity constraints specify the set of achievable aggregate vectors $y$ that can result from block-wise or ballot-wise assignments under resource or voting limitations. Given $m$ rows (e.g., coalition voters or storage blocks) with associated (multiset) score or capacity vectors

$$r^{(v)} = (r^{(v)}_1 \ge r^{(v)}_2 \ge \cdots \ge r^{(v)}_{k'}) \quad \text{for } v=1,\ldots, m,$$

the per-row prefix sums

$R^{(v)}_t = \sum_{i=1}^{t} r^{(v)}_i$

and the aggregate prefix sums

$F(t) = \sum_{v=1}^m R^{(v)}_t$

(for $t=1,\ldots, k'$) define the total capacity up to each block level.

The feasible envelope of real aggregate vectors $y \in \mathbb{R}^{k'}$ is the set satisfying:

• (i) $\sum_{i=1}^t y_i^\downarrow \le F(t)$ for all $t=1,\ldots, k'-1$,
• (ii) $\sum_{i=1}^{k'} y_i = F(k')$,

where $y^\downarrow$ denotes the nonincreasing rearrangement of $y$. For integer realizations with AP (arithmetic progression) block scores of step $g$, a coordinatewise congruence $y_i \equiv L_\text{tot} \pmod{g}$ for all $i$ is further required, where $L_\text{tot}$ aggregates the ballot-wise offsets.

2. Contexts of Application

Block--HLP prefix-sum capacity constraints appear in at least two domains:

• Coalition Power in Multiwinner Elections: In the displacement problem under positional scoring, the constraints determine whether a coalition of size $m$ can guarantee the displacement of $k'$ candidates via coordinated ballot permutations, subject to the structure of ballot scoring ladders (e.g., Borda, $k$-approval) (Guo et al., 23 Jan 2026).
• Prefix Sums in Block-Probe Data Structures: In the cell-probe model for data structures answering prefix-sum queries via a bounded number $q$ of cell accesses, the prefix-sum and block-majorization constraints underpin lower bounds on the space redundancy required for non-adaptive query schemes (0906.1370).

This connection highlights that both voting manipulation and efficient data structure design are controlled by the geometry of feasible prefix-sum vectors under block constraints.

3. The AP–Ladder Majorization–Lattice Theorem

In the setting where each ballot or block provides scores forming an arithmetic progression with constant step $g$, the characterization of integer-feasible vectors is governed by the Majorization–Lattice Theorem. Specifically, an integer vector $y \in \mathbb{Z}^{k'}$ is realizable via permissible permutations if and only if:

• (A) $\sum_{i=1}^t y_i^\downarrow \le F(t)$ for all $t=1,\dots,k'-1$,
• (B) $\sum_{i=1}^{k'} y_i = F(k')$,
• (C) $y_i \equiv L_\text{tot} \pmod{g}$ for all $i$.

Necessity follows from classical majorization combined with the arithmetic progression residue class for each coordinate. Sufficiency is established by reducing the problem to unit-step majorization via translation and scaling, then invoking multilevel Ferrers-matrix decomposition techniques (e.g., Gale–Ryser and Brualdi–Hwang).

For $g=1$ (e.g., Borda), condition (C) is vacuous and all integer points in the Block--HLP polytope are achievable via ballot permutations. For $g>1$, only the integer lattice vectors satisfying the congruence are realizable; the rest are structural false positives of the continuous relaxation (Guo et al., 23 Jan 2026).

4. Feasibility Testing and Computational Methods

The feasibility of a demand vector (e.g., a proposed boost or suppress assignment in an election, or a prefix-sum query in a data structure) can be verified using a sorting-and-prefix-sum approach:

• Adjust the proposed vector to the correct congruence class modulo $g$ (by rounding up for “boost” or down for “suppress” scenarios).
• Sort the adjusted demands in nonincreasing order: $O(k' \log k')$.
• Compute running prefix sums and verify all prefix constraints $s_t \le F(t)$ for $t \le k'$ (plus total sum or equality as needed).
• For $g=1$, all integer vectors passing the prefix test are realizable; for $g>1$, this test filters out infeasible congruence classes.

The overall feasibility oracle operates in $O(k' \log k')$ time. This strong computational efficiency enables scaling to massive instances—a coalition problem on $10^9$ candidates is processed in under 28 seconds, subject to memory constraints (Guo et al., 23 Jan 2026).

5. Distinctions Between Continuous and Integer Integrality

The Block--HLP prefix-sum polytope encompasses all real vectors $y$ satisfying the prefix-sum constraints and total sum equality. However, only specific integer points within this polytope, namely those satisfying the additional congruence (C) in the AP case, correspond to feasible aggregate score assignments.

For instance, if blocks have scores $(8, 5, 2)$ with $g=3$ and $m=3$, the aggregate prefix capacities could be $F(1)=19$, $F(2)=30$, $F(3)=33$. The integer vector $y=(18,11,4)$ satisfies the real Block--HLP constraints but fails the congruence requirement for the residue class of $L_\text{tot}$. This demonstrates that naive (continuous) prefix-sum checks admit false positives when $g>1$; the congruence cuts out a strict sublattice, giving a finer, exact description of feasibility (Guo et al., 23 Jan 2026).

6. Significance in Coalition Power and Data Structure Capacity

In multiwinner Top-$k$ elections, the Block--HLP constraints capture the structure of coalition power. The prefix-sum inequalities (A–B) encode the coalition's collective ability to boost outsider candidates or suppress weak winners, under the positional scoring rule’s per-ballot capacity limits. The congruence (C) accounts for the discrete, step-wise nature of common scoring ladders, producing sharp, exact cutoffs for manipulative feasibility (Guo et al., 23 Jan 2026).

In the block-probe cell-probe model, analogous prefix-sum capacity constraints underpin lower bounds on redundancy: to maintain correct monotonicity and independence across prefix-sum queries, block-probe data structures must allocate additional storage space $r = \Omega(n / \log^{Aq} n)$, with matching upper bounds via non-adaptive constructions (0906.1370). This yields a tight trade-off between probe count $q$ and space redundancy, and the analysis leverages separator, conditional entropy, and majorization-style arguments.

7. Broader Implications and Open Questions

The Block--HLP prefix-sum capacity model underlies both the combinatorics of coalition manipulation and the information-theoretic limits of data structure design. There remains active inquiry into sharp extensions for adaptive probe models, more general scoring rules, and other fundamental query tasks (such as set membership) where block-structured prefix-sum constraints may play a fundamental role (0906.1370). In election theory, further exploration includes the experimental validation of majorization-lattice boundaries and the impact of diminishing marginal returns on coalition efficacy in large-scale real-world social choice datasets (Guo et al., 23 Jan 2026).