Block Rearrangement Problem (BRaP)

Updated 5 September 2025

BRaP is a combinatorial problem that involves transforming an initial block configuration into a target arrangement within constrained settings like grids, stacks, or networks.
It focuses on optimizing cost functions such as move count, makespan, or energy, and is pivotal in domains like warehouse logistics, robotics, and genomics.
Algorithmic strategies include graph search, MAPF, local search, and integer programming, providing robust methods for practical applications and deeper theoretical insights.

The Block Rearrangement Problem (BRaP) is a class of combinatorial and algorithmic problems that focus on the systematic rearrangement of discrete objects—called blocks—within highly constrained environments such as dense grids, stacks, permutations, or networks. BRaP arises naturally in warehouse logistics, robotics, genome rearrangement, container terminals, and memory management. Across its various formulations, the canonical goal is to compute a sequence of actions that will rearrange a given initial configuration of blocks into a prescribed target configuration while optimizing cost functions such as the number of moves, makespan, or energy. Research in this area synthesizes discrete math, graph theory, algorithms, computational genomics, robotic manipulation, combinatorial optimization, and artificial intelligence.

1. Formal Definitions and Problem Frameworks

At its core, BRaP involves a mapping from an initial configuration of a set of blocks within a finite domain (grid, stack, string, or graph) to a target configuration, under a set of allowed moves and constraints. In grid-based warehousing or storage, the environment is typically represented as an undirected graph $G = (V, E)$ , with blocks initially occupying a subset of vertices, and both target and non-target (obstacle) blocks distinguished. The formal definition in a dense warehouse setting is as follows (Fu et al., 31 Aug 2025):

There are target blocks $I = \{1, ..., n_I\}$ , each with a starting vertex $s_i \in V$ and a set of goal vertices $V_i \subseteq V$ .
There are $n_J$ non-target, movable obstacle blocks $J = \{1, ..., n_J\}$ .
Actions $A$ consist of $\operatorname{move}(i, t, u, v)$ for moving block $i$ from $u$ to adjacent and free $v$ , $\operatorname{wait}(i, t)$ , and $\operatorname{complete}(i, t, v)$ when a target reaches a goal.
Solutions are sets of action sequences $P = \{p_i: i \in I \cup J\}$ subject to vertex, edge, and following conflict-avoidance constraints.

The cost of a sequence is

$c(p_i) = \sum_k c(a_i^k)$

for each block $i$ , with solution cost either the composite cost $c(P) = \sum_i c(p_i)$ or the makespan $c_m(P) = \max_i c(p_i)$ , depending on application (Fu et al., 31 Aug 2025).

In stack or container relocation systems, BRaP generalizes to the rearrangement of containers according to specified priority or final stack order, often addressing minimal reshuffling under LIFO queue restrictions (Feillet et al., 2018, Bacci et al., 2022).

In permutation-based genomic contexts, BRaP is cast as the minimum sequence of allowed block operations (transpositions, reversals, prefix block interchanges) needed to transform one permutation into another, subject to biological constraints (Cerbai et al., 2018, Rahman et al., 2019, Nair, 2022, Alexandrino, 27 Apr 2024).

2. Complexity Results and Hardness

The computational complexity of BRaP and related problems is sharply dependent on the problem model, the allowed moves, and structural constraints. Some core results include:

BRaP is NP-hard in general, as established via reductions from well-known partitioning and packing problems, even when only single-block moves between bins (subject to capacity) are allowed (Kam et al., 9 May 2024). In stack-based environments, the problem remains NP-hard for both retrieval and desired rearrangement objectives (Feillet et al., 2018, Bacci et al., 2022).
For grid and permutation models:
- The basic block rearrangement in dense grids with obstacles is NP-hard due to the exponential size of the configuration space $|V|^{|I\cup J|}$ (Fu et al., 31 Aug 2025).
- Sorting permutations by genome-scale rearrangements—transpositions, reversals, and mixtures—is NP-hard for all standard models, including several with weighted cost functions under realistic ratios (Alexandrino, 27 Apr 2024).
- Counting the number of most parsimonious rearrangement scenarios in Single Cut-and-Join models is #P-complete (Bailey et al., 2023).
- Alignment of two partial orders to minimize breakpoints or maximize adjacencies is APX-hard, holding even for restricted input classes (Jiang et al., 2021).

However, certain problem regimes allow for polynomial or better space-time complexity:

The median problem under the multichromosomal circular breakpoint model admits a polynomial-time solution via reduction to maximum non-bipartite matching (Kovac, 2011).
For the Single Cut or Join model, the median counting (#Median) is logspace-computable (FL) (Bailey et al., 2023).
Special structural cases—such as repacking with small items and ample slack, or with power-of-two sizes—allow efficient, sometimes linear-time algorithms (Kam et al., 9 May 2024).

3. Core Algorithmic Techniques

A wide spectrum of algorithmic paradigms have been applied to BRaP:

Graph Search and Configuration-Space Planning: The block configuration is represented as a node in a state-space graph; A*, Dijkstra, joint state-space planners, and domain-specific heuristics (e.g., least-blocking paths) are used to plan sequences of moves (Fu et al., 31 Aug 2025).
Multi-Agent Path Finding (MAPF): Transforming BRaP into a special case of MAPF, MAPF-based planners (LaCAM, PIBT) dynamically resolve block interference and optimize plans for composite cost or makespan (Fu et al., 31 Aug 2025).
Heuristics and Local Search: Both constructive approaches (e.g., greedy selection based on breakpoints or "potential") and improvement heuristics (e.g., dynamic programming-based local search for relocation optimization) can drastically improve practical performance and solution quality (Feillet et al., 2018, Alexandrino, 27 Apr 2024).
Permutation Pattern Theory: For genome rearrangement, the use of generating permutations, basis characterizations, and explicit pattern-avoidance classes form the basis for efficient enumeration and sorting by block operations (Cerbai et al., 2018).
Planning with Expert Heuristics: Domain-specific strategies, such as grouping nested pushes for nonprehensile manipulation (Song et al., 2019), or rearrangement pebble graphs with precomputed roadmaps (Krontiris et al., 2014), exploit combinatorial or physical symmetries to avoid exhaustive exploration.
Linear and Integer Programming: Reconfiguration of multisets with capacity constraints is reduced to ILP or transshipment problems, e.g., Partition ILP models for generalized repacking (Kam et al., 9 May 2024).
Dynamic Programming: State-space DP is employed in local improvement of container relocation and similar "trajectory" optimizations (Feillet et al., 2018).
Approximation Algorithms: Refined analyses for sorting by transpositions yield 1.375-approximation with improved complexity, while breakpoint-based and labeled cycle graph methods yield 2- and 3-approximation algorithms for NP-hard rearrangement distance problems (including those with indels) (Alexandrino, 27 Apr 2024).

4. Structural and Theoretical Insights

Configuration-Space Exponentiality: The joint configuration space scales as $|V|^{n}$ in grids with $n$ movable blocks, necessitating algorithms that reduce the effective branching factor, leverage admissible heuristics, or exploit solution decomposition (Fu et al., 31 Aug 2025).
Permutation Class Structures: The block transposition and prefix block transposition models lead to permutation balls $B_k^{(d)}$ , completely characterized by generating permutations and polynomially bounded bases for small $k$ , enabling explicit enumeration and pattern-avoidance sorting (Cerbai et al., 2018).
Approximation Hardness: Many restrictions of BRaP, including genome block rearrangements and partial order linearizations, are APX-hard, with no PTAS expected for generic settings (Jiang et al., 2021).
Median and Small Phylogeny Dichotomies: The tractability of the median and small phylogeny problems is tightly linked to the structural features of the underlying model: tractable for circular breakpoint genome models, but NP-hard—and sometimes APX-hard—for unichromosomal, multilinear, or DCJ/reversal frameworks (Kovac, 2011).

5. Practical Applications

BRaP presents an underlying combinatorial structure common to several applied domains:

Warehouse Automation and Logistics: Optimizing the movement of product bins, storage containers, or inventory items to facilitate deep access, batch rearrangement, or throughput maximization in dense distribution grids (Fu et al., 31 Aug 2025, Szegedy et al., 2020).
Container Terminals and Yard Management: Determining minimal container relocations or achieving target stacking orders under tight sequencing and space constraints (Feillet et al., 2018, Bacci et al., 2022).
Genome Rearrangement in Computational Biology: Estimating mutational distances, reconstructing evolutionary scenarios, and modeling genome editing operations in both balanced (permutation) and unbalanced (string with indels) settings (Cerbai et al., 2018, Alexandrino, 27 Apr 2024, Bailey et al., 2023).
Robotics and Manipulation Planning: Manipulation of multiple objects on confined or cluttered surfaces, with solutions spanning from rearrangement pebble graphs to kinodynamic planning with uncertainty handling (Krontiris et al., 2014, Song et al., 2019, Ren et al., 2023, Bayraktar et al., 2022).
Data Center Resource Reallocation: Repacking virtual machines among physical servers with capacity and feasibility constraints modeled as repacking/reconfiguration problems (Kam et al., 9 May 2024).

6. Comparative Empirical Findings and Performance

Extensive benchmark testing across models and environments has yielded the following salient results:

MAPF-based and heuristic search algorithms for dense grids can generate high-quality plans (in composite cost and makespan), scaling to large grids (up to 80×80) with high success rates (>99%) and sub-second planning per instance in many cases (Fu et al., 31 Aug 2025).
Approximation algorithms based on refined structural analysis (cycle graphs, breakpoint accounting) achieve provable 2- and 3-approximation guarantees while significantly outperforming naïve or purely constructive heuristics—for instance, achieving up to 50% fewer relocations versus prior methods in large block relocation benchmarks (Feillet et al., 2018, Alexandrino, 27 Apr 2024).
For reconfiguration of multisets (general block repacking), specialized algorithms handle the small-item, powers-of-two, and partitioned cases efficiently, while the general case remains NP-hard (Kam et al., 9 May 2024).
In experimental comparisons for sorting by transpositions, recent algorithms (ALG3) achieve the 1.375-approximation theoretically while producing optimal solutions in at least 88% of instances tested up to $n=12$ (Alexandrino, 27 Apr 2024).

7. Open Challenges and Future Research Directions

Several open questions and avenues for future research are highlighted:

Handling Exponential Search Spaces: Continued progress in heuristics, decomposition, symbolic planning, and parallel search is needed to manage the combinatorial explosion in high-density settings.
Algorithmic Hardness and Approximability: Determining tight complexity and approximability boundaries—particularly for generalized reconfiguration, weighted or signed block problems, and hybrid models mixing operations—is a persistent theoretical challenge (Alexandrino, 27 Apr 2024, Jiang et al., 2021).
Resource-Optimal Strategies: Investigating not merely move-minimal solutions, but those that minimize temporary resource requirements (e.g., buffer space or auxiliary workspace in reconfiguration) is an emerging thread (Kam et al., 9 May 2024).
Integration of Learning and Planning: Real-time adaptive or learning-augmented heuristic methods for BRaP in dynamic, uncertain, or partially observed environments is a promising frontier, leveraging progress in kinodynamic planning under uncertainty (Ren et al., 2023).
Cross-Domain Transfer: Applying theoretical tools—such as graph-theoretical matching reductions, ILP/transshipment formulations, or permutation class enumeration—to new classes of real-world rearrangement problems, including power supply networks or distributed data shuffling (Kam et al., 9 May 2024).
Benchmarking and Instance Generation: Expansion of public, large-scale benchmark instances (such as LBRI or bin-packing repacking datasets) will be vital for systematically evaluating and comparing emerging methods across the spectrum of BRaP applications (Bacci et al., 2022, Kam et al., 9 May 2024).

BRaP thus serves as a central, multidisciplinary motif linking foundational combinatorial optimization with modern applications in automation, bioinformatics, and resource management. Advances in this area will continue to be shaped by both the nuanced structural properties of specific domains and the general algorithmic principles of graph search, combinatorial design, and constraint satisfaction.