Tabletop Object Rearrangement: Structure, Complexity, and Efficient Combinatorial Search-Based Solutions

Kai Gao

REVIEW 4 major objections 5 minor 120 references

Tabletop Object Rearrangement: Structure, Complexity, and Efficient Combinatorial Search-Based Solutions

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Computing the minimum running buffer for tabletop rearrangement is NP-hard, yet exact search can handle over 100 objects.

desk verdict The TORE running-buffer chapter is genuinely solid; the rest is a compilation of the author's previously published work, with the usual dissertation-level reproducibility gaps. read the letter →

arxiv 2412.15398 v2 pith:MTVKC7NX submitted 2024-12-19 cs.RO

Kai Gao This is my paper · ORCID

classification cs.RO MSC 68T4068Q1705C85

keywords tabletopobjectrearrangementrunningbufferminimizationdependencygraphNP-hardnesslazyverificationdepth-firstdynamicprogrammingdual-armcoordinationmotionplanning

topics P versus NP

open problems P versus NP

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This thesis argues that the minimum number of temporary storage slots ("running buffers") needed to rearrange objects on a tabletop is governed entirely by the ordering of objects in a dependency graph built from start and goal poses. It proves that computing this minimum is NP-hard, that for uniform cylinders the required buffer count can grow to about the square root of the number of objects, and that this growth rate is the best possible bound for unlabeled objects. On the practical side, it develops exact depth-first dynamic-programming searches that compute optimal buffer plans for instances with over a hundred densely packed objects, and a "lazy buffer verification" strategy that turns those plans into collision-free moves inside the workspace. The same machinery extends to dual-arm coordination and mobile-base rearrangement. A sympathetic reader would care because buffer space is the scarce resource in cluttered rearrangement, and the paper shows both why it is provably hard to minimize and that exact minimization is nonetheless computationally accessible at realistic scales.

What carries the argument

The dependency graph is the central object: a directed graph whose arcs record that object i occupies object j's goal pose in the labeled setting, and a planar bipartite graph between start and goal poses in the unlabeled setting. The paper's key identity is that MRB equals the minimum over linear vertex orderings of the number of vertices whose dependency edges cross the moving boundary (vertex separation), up to an additive-one slack; this connects running buffers to vertex separation, links NP-hardness to planar vertex separation, and lets separator-based recursion and depth-first dynamic programming carry the algorithms.

What would settle it

A concrete TORE instance whose optimal plan moves an object to a buffer more than once before reaching its goal, achieving a smaller running-buffer peak than any single linear vertex ordering of the dependency graph, would refute the claimed reduction of MRB to linear ordering. Constructing such an instance is the direct test of Proposition 2.3.1 and Theorem 2.4.1.

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Extended reading notes

Core claim

The paper's central claim is that the running-buffer problem for tabletop object rearrangement with overhand grasps (TORO) reduces to a graph-ordering problem: every rearrangement plan corresponds to a linear ordering of the vertices of a labeled or unlabeled dependency graph, and the maximum number of objects stored simultaneously equals the "cut" size of that ordering. Computing the minimum such cut (MRB) is NP-hard even when only collision-free placement between any two poses is assumed, so no polynomial-time optimal algorithm exists unless P = NP. The paper also establishes that MRB can be as large as Ω(√n) for n uniform cylinders in both labeled and unlabeled settings, and that the unlabeled bound is tight because a separator-based algorithm always produces plans with O(√n) running buffers. For the labeled setting the gap between lower and upper bounds is left open. It then shows that exact depth-first dynamic programming finds MRB-optimal plans and witness sequences for random and adversarially constructed instances with over 100 objects, and that the same planning primitives, combined with lazy buffer verification, produce state-of-the-art solutions for internal-buffer, heterogeneous-object, dual-arm, and mobile-robot variants.

Load-bearing premise

The dependency-graph abstraction assumes that any object can be moved from any collision-free pose to any other by an overhand pick-n-place, so pairwise footprint overlap is the only constraint that determines whether one object blocks another.

Editorial extensions

If this is right

For any TORE instance, the minimum number of external buffer slots needed for feasibility is exactly the MRB of its dependency graph, so hardware designers can size buffer space from graph structure.
Unlabeled instances of uniform cylinders are always solvable with O(√n) running buffers, and that bound cannot be improved in the worst case.
Exact MRB computation is NP-hard, so large arbitrary instances cannot be solved exactly in polynomial time unless P = NP.
Depth-first dynamic programming (DFDP) computes optimal MRB and witness plans for instances with over 100 objects, and can also minimize total buffers subject to the MRB constraint.
Lazy buffer verification converts these external-buffer plans into feasible internal-buffer plans reliably and up to two orders of magnitude faster than prior baselines.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The linear-ordering reduction suggests a direct connection to graph layout problems such as pathwidth and search number; an exact polynomial algorithm for bounded-treewidth dependency graphs is a natural next step that follows from existing separator structure.
Because random instances have MRB values near zero or a small constant, practical systems can probabilistically rely on a small constant number of buffer slots; a testable extension is to characterize the distribution of MRB under random geometric placements.
The lazy buffer verification idea could be applied to other task-and-motion planning settings where a coarse combinatorial schedule is refined into geometric feasibility, such as rearrangement with pushing actions or with deformable objects.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Kai, quick read of the Gao dissertation (arXiv:2412.15398). The thing to know: the TORE chapter (running buffer minimization) is the real meat, and it holds up. The rest is a faithful compilation of the author's own prior papers—useful as a thesis, not as new material.

What's new and good: the reduction from vertex separation to MRB (Lemma 2.4.1, Theorem 2.4.1) is clean and correct in outline. The NP-hardness for TORE via planar graph gadgets is plausible; I'd want to see the gadget details, but they are in the appendix. The Omega(sqrt n) lower bound via dependency grids and the matching O(sqrt n) separator-based algorithm for the unlabeled case are genuinely nice: they give the first tight bounds I know for uniform cylinders. The exact DP/DFDP algorithms are not conceptually deep but scale to 100 objects, which is more than prior practical limits. I don't see a load-bearing flaw in the central TORE argument.

Where it gets softer. First, several load-bearing proofs (Lemma 2.5.1, Theorem 2.5.2, Propositions 2.3.2/2.3.3) are deferred to Appendix A and given only as sketches in the main text. In the arXiv version, the appendix is not even included, so an external reader cannot verify them. That is fixable, but it is a real reproducibility gap. Second, no code or data is released. The experimental claims—exactness, success rates, speedups—are not independently checkable. Given that the dissertation explicitly compiles P1-P7, some of these experiments are already published, but the arXiv artifact itself does not ship them. Third, the ORLA* chapter (Chapter 5) is truncated in the copy under review; I can't assess it fully. Fourth, the TORI lazy buffer allocation (Algorithms 6-7) is heuristic: the greedy buffer generation with random restarts has no completeness theorem. That doesn't undermine the TORE results, but it means the TORI/dual-arm chapters should be read as engineering contributions with empirical support, not as guaranteed-solve planners.

The citation pattern is fine: self-citation is appropriate here because the results are the author's own and are independently published. No circularity red flag.

Who this is for: someone working on rearrangement planning, especially the buffer-space literature, will get value from the TORE chapter. The other chapters are useful as a consolidated reference to the author's TRLB/MODAP/ORLA* line.

My recommendation: send it to a serious referee, but the referee should be told to focus on the TORE structural results and to request the appendix proofs and a code/data release before acceptance. Would I cite it? Yes, for the MRB bounds and the DFDP algorithm. For a reading group, I'd assign only Chapter 2.

Referee Report

4 major / 5 minor

Summary. The dissertation studies tabletop object rearrangement with overhand grasps (TORO) under four settings: external buffers (TORE), internal buffers (TORI), dual-arm coordination (CDR), and mobile-robot rearrangement (ORLA*). The central theoretical contribution is a structural analysis of the minimum running buffer (MRB) problem for TORE. The thesis proves that computing MRB is NP-hard (Theorems 2.4.1 and 2.4.2), establishes an Omega(sqrt(n)) lower bound on MRB for uniform cylinders in both labeled and unlabeled settings, provides a matching O(sqrt(n)) upper bound for the unlabeled setting via a separator-based algorithm (SEPPLAN), and develops exact search algorithms (DP, DFDP, PQS) that scale to over 100 objects on random and structured instances. The later chapters extend these ideas to internal-buffer rearrangement with lazy buffer allocation (TRLB), dual-arm task and motion planning (MCHS/MODAP), and A*-based lazy buffer allocation on mobile platforms (ORLA*). The manuscript is a PhD thesis and largely consolidates the author's previously published work.

Significance. If the TORE results are fully rigorous, they provide a valuable bridge between rearrangement planning and graph layout problems: the running-buffer objective is connected to vertex separation, the dependency graph abstraction is shown to be the right combinatorial object for overhand-grasp manipulation with external buffers, and the exact algorithms with witnessed plans are practically useful and parameter-free. The empirical evaluation is extensive, including physical robot demonstrations, and the DFDP algorithm's scaling to over a hundred objects is genuinely impressive. The later chapters are more heuristic in nature but the experimental methodology is generally sound. The thesis does not overclaim exactness for the TORI/TRLB framework, framing it as a high-quality heuristic, which is appropriate. The main risk is not circularity or fitted constants but the completeness of several load-bearing proofs, which are deferred to the appendix or stated informally.

major comments (4)

[Section 2.4.2 (around the equation MRB(G^l) = min_phi RB(G^l, phi))] The equivalence between arbitrary rearrangement plans and linear orderings of the dependency graph is asserted as 'straightforward to see' but is not proved. A formal argument must show that for any plan using b running buffers there exists a linear ordering phi with RB(G^l, phi) <= b, and conversely that the greedy execution of any ordering uses exactly the computed number of running buffers. The forward direction is the nontrivial part: one must prove that an optimal plan can be assumed to move each object from its start pose at most once, either directly to its goal or to a buffer, and that buffered objects are placed at their goals at the earliest possible time, without increasing the maximum running-buffer count. This equivalence is load-bearing because the DP in Algorithm 2 and the hardness reduction in Theorem 2.4.1 both search only over such restricted linear-order plans.
[Section 2.4.3, Theorem 2.4.2 and Figure 2.7] The reduction from planar vertex separation to TORE is described informally with 'object gadgets' and a rotation by 'some small angle epsilon'. To constitute a valid NP-hardness proof for TORE instances (as opposed to arbitrary dependency graphs), the construction must specify the exact object footprint geometry, the epsilon value, and a proof that the resulting labeled dependency graph contains exactly the bidirectional edges of the original planar graph with no extraneous dependencies between non-adjacent gadgets. The sentence 'It is straightforward to check' is not sufficient for a complexity-theoretic claim, especially because the construction must work for all planar graphs, including those with vertices of degree up to five.
[Section 2.5.2, Theorem 2.5.2 and the SEPPLAN sketch] The main text states 'As shown in detail in the appendix, SEPPLAN guarantees an MRB upper bound of 20/(1-sqrt(2/3)) sqrt(n)' but the derivation of this constant and the proof of the key invariant (that after removing the separator C and the neighbors of goal vertices in C, the subgraphs A' and B' are independent and can be solved recursively without exceeding the claimed bound) are not presented in the main text. If the appendix contains a complete proof, the manuscript should refer to it explicitly; if not, this is a gap in a central upper-bound theorem. The proof of Lemma 2.5.1 is similarly deferred, which is acceptable for a thesis but should be flagged in a journal version.
[Section 3.4, Algorithms 6 and 7 (TRLB)] The TRLB framework's two-step procedure, which computes a primitive plan using the external-buffer abstraction and then greedily allocates internal buffers with random restarts, has no completeness guarantee. The manuscript states that randomness allows recovery from failures, but there is no theorem ensuring that a feasible internal-buffer plan will be found when one exists, nor a bound on the number of restarts. The chapter is consequently a heuristic contribution, which is acceptable given the experimental evidence, but the limitation should be stated explicitly in Section 3.4 and in the chapter summary to avoid the impression that TRLB is an exact planner.

minor comments (5)

[Abstract and Chapter 1] The abstract and Chapter 1 promise 'efficient combinatorial search-based solutions' and 'exact algorithms' for TORE, but the later chapters (TORI, CDR, ORLA*) are largely heuristic or approximate; a brief sentence clarifying which parts are exact and which are heuristic would help the reader.
[Section 2.7.1 (experimental setup)] Gurobi is cited as 'Gurobi 9.16.0', which does not match the known version numbering (Gurobi 9.1.x is a plausible intended version); please verify the version number.
[Appendix A, Figure A.2 caption] The caption contains a duplicated phrase: 'A counterexample of the proof of Prop.Proposition 2.3.3' should read 'A counterexample to the proof of Proposition 2.3.3'.
[Chapter 4 (notation)] The density measure is denoted rho in Chapter 2 and D in Chapter 4; using consistent notation across chapters would reduce confusion, even though the definitions are analogous.
[General (reproducibility)] The empirical results would be easier to reproduce if the code and benchmark instance generators were made available; the thesis currently provides only textual descriptions and videos.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the MRB hardness and bound theorems are proven in-text from external standard results; self-citations to the author's prior TORO/TRLB papers are background support, not load-bearing circular steps.

full rationale

Walked the claimed derivation chain. The load-bearing structural results—MRB NP-hardness (Theorems 2.4.1–2.4.2), the Ω(√n) lower bound for uniform cylinders (Theorem 2.5.1), and its O(√n) separator-based upper bound for URBM (Theorem 2.5.2)—are proven in the thesis by explicit reductions to established external results (vertex-separation inapproximability [55]; VSP NP-completeness on planar bounded-degree graphs [67]; the planar separator theorem [57]; straight-line planar embeddings [68]) and by concrete constructions (object gadgets, dependency grids) that do not presuppose the conclusions. The MRB/linear-ordering identification (Section 2.4.2, Lemma 2.4.1) is an argued equivalence between the Problem 1 minimization over plans and a combinatorial ordering objective, and the DP/DFDP/PQS recurrences compute that objective exactly; no fitted parameter is renamed as a prediction, and the identity max(0,|N(g)|−|g|) in Section 2.5.2 is a displacement-conservation count under the move semantics, not a circular definition. Self-citations are pervasive because this is a compilation thesis of the author's publications P1–P7: [1] supplies the TORO model and the total-buffer/MFVS equivalence, while [74], [81], [97] supply running-buffer, MCTS, and lazy-buffer background. These are published, parameter-free results and are not the load-bearing premise for the new theorems; Chapter 3's 'TORI is also NP-hard' simply inherits published TORE hardness, and the thesis independently proves TORE MRB-hardness in Chapter 2. StabilNet's fitted weights (Chapter 5) form an empirical feasibility oracle evaluated by ablations and never feed back into the structural theorems. TRLB's greedy internal-buffer allocation with random-restart recovery is explicitly presented as a heuristic (Section 3.4.2), an honest limitation rather than circularity. Comparisons against external baselines (BiRRT(fmRS), MCTS, lattice benchmarks, real-robot execution) are self-contained. Verdict: no circular step; residual concern is only minor, non-load-bearing self-citation, consistent with score 2.

Assumptions & free parameters 3 free parameters · 8 assumptions · 1 invented entities

The theoretical chapters are derivational and contain no fitted constants in the MRB, TORE, or TRLB arguments. The empirical chapters add fitted components, chiefly StabilNet weights, plus hand-set simulation cost models and sampling constants. The axioms are mostly standard graph-theoretic tools and the explicitly stated workspace and grasp models.

free parameters (3)

StabilNet network weights = trained on simulated stability data
ORLA* uses learning-based stability predictions for multi-layered placements; these weights are fitted to data, and their accuracy affects the general-shaped-object success claims in Section 5.5.
CDR and FC cost parameters tg, tr, th, td = set equal to td in experiments
The FC makespan evaluation in Chapter 4 chooses all manipulation times equal to one traversal time, a hand-set model that influences comparative execution-time claims but not the structural theorems.
MODAP sampling constants gamma0, gamma_n, k1, k2 = constants used in Algorithms 11 and 12
These hand-chosen constants affect collision-free dual-arm goal sampling and path-yielding behavior, and therefore affect the empirical MODAP results in Section 4.6.

assumptions (8)

standard math Planar separator theorem of Lipton and Tarjan
Used in Algorithm 1 (SEPPLAN) in Section 2.5.2 to split the planar unlabeled dependency graph into independent subgraphs.
standard math NP-hardness and non-approximability of vertex separation
Used in Theorems 2.4.1 and 2.4.2 to transfer hardness to MRB computation; relies on results of [55] and [67].
standard math Fáry's theorem that every planar graph has a straight-line embedding
Used in the TORE NP-hardness gadget construction in Section 2.4.3 to convert a planar graph drawing into object poses.
domain assumption Overhand pick-n-place can realize any sequence of collision-free poses
Prop 2.3.1 states that the labeled dependency graph fully captures the TORE problem, an assumption that abstracts away kinematics, trajectory feasibility, and gripper constraints.
domain assumption Unlabeled dependency graphs for uniform discs are planar with maximum degree 5
Proposition 2.3.3, proved in Appendix A, is the geometric bound that makes the O(√n) separator-based upper bound in Theorem 2.5.2 valid.
domain assumption Greedy buffer allocation with random restarts is sufficient for TORI
Algorithms 6 and 7 in Section 3.4 provide empirical recovery from buffer allocation failures, but no completeness or bounded-suboptimality proof is provided.
domain assumption Dual-arm reachability S(r1) union S(r2) covers the workspace and handoff poses are always available
The CDR model in Section 4.3.1 assumes task-level scheduling can ignore low-level geometry, including robot-arm collision details.
standard math NP-hardness of feedback vertex set and TSP as established in [1]
Cited in Sections 2.2 and 4.2 as background for why TORO action sequencing is NP-hard.

invented entities (1)

StabilNet independent evidence
purpose: Learned stability predictor used by ORLA* for multi-layered placements in mobile-robot tabletop rearrangement (Section 5.4 and 5.5).
StabilNet makes falsifiable predictions about physical placement stability that can be checked in simulation or on real robots, although it is trained on data rather than derived from physics.

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Cite this review

Pith. "Pith review of Tabletop Object Rearrangement: Structure, Complexity, and Efficient Combinatorial Search-Based Solutions." pith.science (2026). https://pith.science/paper/MTVKC7NX

@misc{pith2026241215398,
  author       = {Pith},
  title        = {Pith review of: Tabletop Object Rearrangement: Structure, Complexity, and Efficient Combinatorial Search-Based Solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MTVKC7NX}},
  note         = {Machine review of arXiv:2412.15398}
}

read the original abstract

This thesis provides an in-depth structural analysis and efficient algorithmic solutions for tabletop object rearrangement with overhand grasps (TORO), a foundational task in advancing intelligent robotic manipulation. Rearranging multiple objects in a confined workspace presents two primary challenges: sequencing actions to minimize pick-and-place operations - an NP-hard problem in TORO - and determining temporary object placements ("buffer poses") within a cluttered environment, which is essential yet highly complex. For TORO with available external free space, this work investigates the minimum buffer space, or "running buffer size," required for temporary relocations, presenting both theoretical insights and exact algorithms. For TORO without external free space, the concept of lazy buffer verification is introduced, with its efficiency evaluated across various manipulator configurations, including single-arm, dual-arm, and mobile manipulators.

Figures

Figures reproduced from arXiv: 2412.15398 by the authors.

**Figure 3.** [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

Figures from the paper (129 more)

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p022_1.png]

**Figure 1.1.** Figure 1.1: A TORO instance where the three soda cans are to be rearranged from the left configuration to the right configuration [PITH_FULL_IMAGE:figures/full_fig_p022_1_1.png]

**Figure 1.2.** Figure 1.2: [Left] PyBullet setup for the Cooperative Multi-Robot Rearrangement problem, [PITH_FULL_IMAGE:figures/full_fig_p025_1_2.png]

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p025_1.png]

**Figure 1.3.** Figure 1.3: An example of the Mobile Robot Tabletop Rearrangement (MoTaR) setup. [PITH_FULL_IMAGE:figures/full_fig_p026_1_3.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p030_2.png]

**Figure 2.1.** Figure 2.1: A 7-object labeled instance with uniform cylinders; we will use this instance [PITH_FULL_IMAGE:figures/full_fig_p031_2_1.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p031_2.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p032_2.png]

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p032_1.png]

**Figure 2.2.** Figure 2.2: Two configurations of the setup given in [PITH_FULL_IMAGE:figures/full_fig_p033_2_2.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p033_2.png]

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p035_1.png]

**Figure 2.3.** Figure 2.3: An instance with the labeled dependency graph formed by 3 copies of [PITH_FULL_IMAGE:figures/full_fig_p036_2_3.png]

**Figure 2.4.** Figure 2.4: An LRBM instance with uniform thin cuboids (left) and its labeled dependency graph, where the total number of buffers needed is more than the size of the MFVS when the number of running buffers is minimized. its goal pose if it has no further dependencies; otherwise,…

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p037_2.png]

**Figure 2.5.** Figure 2.5: Two linear orderings of vertices of the labeled dependency graph from [PITH_FULL_IMAGE:figures/full_fig_p038_2_5.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p038_2.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p039_2.png]

**Figure 2.6.** Figure 2.6: (a) An undirected graph and a linear ordering of its vertices. (b) A correspond [PITH_FULL_IMAGE:figures/full_fig_p039_2_6.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p040_2.png]

**Figure 2.7.** Figure 2.7: (a) An undirected planar graph with five vertices (b) “Object gadgets” for two [PITH_FULL_IMAGE:figures/full_fig_p041_2_7.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p041_2.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p042_2.png]

**Figure 2.8.** Figure 2.8: An instance with 6 cuboids where horizontal and vertical sets represent start and [PITH_FULL_IMAGE:figures/full_fig_p042_2_8.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p043_2.png]

**Figure 2.9.** Figure 2.9: A URBM instance (left) and its unlabeled dependency graph (right), a 4 × 3 dependency grid. Unshaded (resp., shaded) discs/vertices indicate start poses (resp., goal poses). Because URBM instances always have lower or equal MRB than the LRBM instances with the same o…

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p044_2.png]

**Figure 2.10.** Figure 2.10: An example of a 9-object LRBM yielding Ω(√ n) MRB (left) and the corresponding dependency graph (right) [PITH_FULL_IMAGE:figures/full_fig_p044_2_10.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p045_2.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p046_2.png]

**Figure 2.11.** Figure 2.11: The recursive solver SEPPLAN for URBM. (a) A O( p |V |) vertex separator for the planar dependency graph. (b) By removing the start vertices in C and the neighbors of the goal vertices in C, the remaining graph consists of two independent subgraphs and isolated goal…

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p048_2.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p050_2.png]

**Figure 2.12.** Figure 2.12: Generating dense instances using a physics-engine-based simulator through [PITH_FULL_IMAGE:figures/full_fig_p055_2_12.png]

**Figure 2.13.** Figure 2.13: Unlabeled arrangements with ρ = 0.2, 0.4, 0.6 respectively. From two randomly generated object placements with the same ρ and n values, a URBM instance can be readily created by superimposing one over the other. LRBM instances can be generated from URBM instances by…

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p056_2.png]

**Figure 2.14.** Figure 2.14: Performance of DFDP and DP over LRBM. The top row shows the average computation time (s) and the bottom row the success rate, for density levels ρ = 0.2, 0.3, 0.4, from left to right. The x-axis denotes the number of objects involved in a test case. The actual MRB s…

**Figure 2.15.** Figure 2.15: For LRBM instances with ρ = 0.2-0.4 and n = 20-100, the left figure shows average MRB size and range. The right figure shows the size of the largest connected component of the dependency graph. For LRBM with ρ = 0.3 and n up to 50, we computed the MFVS sizes using T…

**Figure 2.16.** Figure 2.16: (a) Comparison between size of MRB and MFVS. (b) Computation time [PITH_FULL_IMAGE:figures/full_fig_p058_2_16.png]

**Figure 2.17.** Figure 2.17: Performance of DFDP and PQS over URBM. The top row shows the average computation time and the bottom row shows the success rate, for density levels ρ = 0.4, 0.5, 0.6, from left to right. 20 40 60 80 100 0.0 0.2 0.4 0.6 ρ=0.4 ρ=0.5 ρ=0.6 [PITH_FULL_IMAGE:figures/fu…

**Figure 2.18.** Figure 2.18: Average MRB size for URBM instances with ρ = 0.4 − 0.6 and n = 20 − 100. For ρ = 0.4 and 0.5, the MRB sizes are near zero as the number of objects goes beyond 20. time and the corresponding MRB are shown in [PITH_FULL_IMAGE:figures/full_fig_p059_2_18.png]

**Figure 2.19.** Figure 2.19: For handcrafted cases and different numbers of objects, the left figure shows [PITH_FULL_IMAGE:figures/full_fig_p060_2_19.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p060_2.png]

**Figure 2.20.** Figure 2.20: Our hardware setup for executing rearrangement plans computed by proposed [PITH_FULL_IMAGE:figures/full_fig_p061_2_20.png]

**Figure 2.21.** Figure 2.21: An example instance of our experiment. The right side of the pad works as an [PITH_FULL_IMAGE:figures/full_fig_p062_2_21.png]

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p064_1.png]

**Figure 3.1.** Figure 3.1: A robot arm rearranges word patterns with overhand grasps, minimizing the [PITH_FULL_IMAGE:figures/full_fig_p067_3_1.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p068_3.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p070_3.png]

**Figure 3.2.** Figure 3.2: A working example with three objects defined in (a). The primitive plan is [( [PITH_FULL_IMAGE:figures/full_fig_p071_3_2.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p073_3.png]

**Figure 3.3.** Figure 3.3: A four-object example of preprocessing. The green and cyan discs represent [PITH_FULL_IMAGE:figures/full_fig_p073_3_3.png]

**Figure 3.4.** Figure 3.4: [Left] Our hardware setup for evaluating tabletop rearrangement of multiple [PITH_FULL_IMAGE:figures/full_fig_p074_3_4.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p074_3.png]

**Figure 3.5.** Figure 3.5: An illustration of the Minkowski difference [PITH_FULL_IMAGE:figures/full_fig_p077_3_5.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p077_3.png]

**Figure 3.6.** Figure 3.6: [left] 10 cylinders with ρ = 0.5, [right] 10 cuboids with ρ = 0.4 [PITH_FULL_IMAGE:figures/full_fig_p081_3_6.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p082_3.png]

**Figure 3.7.** Figure 3.7: Comparison of primitive planners with 10-40 cylinders and density levels ρ = 0.3 (top), 0.5 (bottom) (left: computation time in seconds; middle: success rate; right: number of actions as multiples of |O|). the environment, and the smaller the configuration space size…

**Figure 3.8.** Figure 3.8: Comparison of buffer allocation methods with [PITH_FULL_IMAGE:figures/full_fig_p083_3_8.png]

**Figure 3.9.** Figure 3.9: Comparison among high-level frameworks and OS with preprocessing. There [PITH_FULL_IMAGE:figures/full_fig_p084_3_9.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p084_3.png]

**Figure 3.10.** Figure 3.10: Comparison between ST and BST frameworks with “dense-small” instances [PITH_FULL_IMAGE:figures/full_fig_p085_3_10.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p085_3.png]

**Figure 3.11.** Figure 3.11: Comparison of algorithms with 20-100 cylinders at density level [PITH_FULL_IMAGE:figures/full_fig_p086_3_11.png]

**Figure 3.12.** Figure 3.12: Comparison of methods on “dense-small” instances where 5-8 objects are [PITH_FULL_IMAGE:figures/full_fig_p086_3_12.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p086_3.png]

**Figure 3.13.** Figure 3.13: Comparison among methods in lattice instances with 3-35 objects. [left] [PITH_FULL_IMAGE:figures/full_fig_p087_3_13.png]

**Figure 3.14.** Figure 3.14: Comparison between methods in cuboid instances with [PITH_FULL_IMAGE:figures/full_fig_p087_3_14.png]

**Figure 3.15.** Figure 3.15: Experimental settings in addition to that of [PITH_FULL_IMAGE:figures/full_fig_p087_3_15.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p088_3.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p089_3.png]

**Figure 3.16.** Figure 3.16: Illustrations of some simulation test cases at different densities. [top] RAND [PITH_FULL_IMAGE:figures/full_fig_p089_3_16.png]

**Figure 3.17.** Figure 3.17: Algorithm performance in RAND instances with [PITH_FULL_IMAGE:figures/full_fig_p090_3_17.png]

**Figure 3.18.** Figure 3.18: Algorithm performance in SQ instances with [PITH_FULL_IMAGE:figures/full_fig_p090_3_18.png]

**Figure 3.19.** Figure 3.19: Algorithm performance in RAND instances under different density levels and [PITH_FULL_IMAGE:figures/full_fig_p091_3_19.png]

**Figure 3.20.** Figure 3.20: Algorithm performance in SQ instances under different density levels and [PITH_FULL_IMAGE:figures/full_fig_p092_3_20.png]

**Figure 3.21.** Figure 3.21: Algorithm performance in RAND instances with TI objective. [PITH_FULL_IMAGE:figures/full_fig_p093_3_21.png]

**Figure 3.22.** Figure 3.22: Algorithm performance in SQ instances with TI objective. [PITH_FULL_IMAGE:figures/full_fig_p093_3_22.png]

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p093_3.png]

**Figure 3.23.** Figure 3.23: Physical experiment/demo setups. [left] Dense SQ instance with [PITH_FULL_IMAGE:figures/full_fig_p094_3_23.png]

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p096_1.png]

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p099_1.png]

**Figure 4.1.** Figure 4.1: [Left] A working example of CDR instance with [PITH_FULL_IMAGE:figures/full_fig_p099_4_1.png]

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p100_1.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p102_4.png]

**Figure 4.2.** Figure 4.2: Decision process of generating possible primitive actions. [Upper] The decision [PITH_FULL_IMAGE:figures/full_fig_p103_4_2.png]

**Figure 4.3.** Figure 4.3: [Left] A path on the arrangement state space for the instance in [PITH_FULL_IMAGE:figures/full_fig_p104_4_3.png]

**Figure 4.4.** Figure 4.4: [Top] A CDR instance that needs a buffer. [Bottom] An rearrangement plan for [PITH_FULL_IMAGE:figures/full_fig_p106_4_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p106_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p107_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p109_4.png]

**Figure 4.5.** Figure 4.5: [left] A configuration of our dual-arm setup, also showing that the camera is [PITH_FULL_IMAGE:figures/full_fig_p114_4_5.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p115_4.png]

**Figure 4.6.** Figure 4.6: A 20-cylinder example of rearrangement instance with [PITH_FULL_IMAGE:figures/full_fig_p115_4_6.png]

**Figure 4.7.** Figure 4.7: Performance of FCHS and MCHS in instances with [PITH_FULL_IMAGE:figures/full_fig_p116_4_7.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p116_4.png]

**Figure 4.8.** Figure 4.8: Algorithm comparison in environments with different density levels [PITH_FULL_IMAGE:figures/full_fig_p117_4_8.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p117_4.png]

**Figure 4.9.** Figure 4.9: Evaluation of instances with different ρ. The x-axis represents the number of objects. Fully Overlapping Workspaces. MCHS can also compute CDR plans with full overlap (CDRF), i.e., ρ = 1.0. As shown in [PITH_FULL_IMAGE:figures/full_fig_p118_4_9.png]

**Figure 4.10.** Figure 4.10: An instance of CDR with full overlap (CDRF), where each arm can reach [PITH_FULL_IMAGE:figures/full_fig_p119_4_10.png]

**Figure 4.11.** Figure 4.11: Algorithm performance in CDR instances with [PITH_FULL_IMAGE:figures/full_fig_p119_4_11.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p120_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p121_4.png]

**Figure 4.12.** Figure 4.12: [top] RMSD of configuration and end-effector position on a sampled trajectory [PITH_FULL_IMAGE:figures/full_fig_p123_4_12.png]

**Figure 4.13.** Figure 4.13: Trajectories of BL, BL-TP, and MODAP for the a typical rearrangement task. Each row, from top to bottom, corresponds to a robot joint in radians from one to six. Each column shows the trajectory of the six joint angles of a robot [PITH_FULL_IMAGE:figures/full_fig_p…

**Figure 4.14.** Figure 4.14: Average execution time (left) and trajectory total length (right) computed by [PITH_FULL_IMAGE:figures/full_fig_p125_4_14.png]

**Figure 4.15.** Figure 4.15: Two examples of start and goal configurations with 15 objects at overlap rates [PITH_FULL_IMAGE:figures/full_fig_p125_4_15.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p129_5.png]

**Figure 5.1.** Figure 5.1: [Left] An example of the EE scenario, where the table is small and the robot [PITH_FULL_IMAGE:figures/full_fig_p130_5_1.png]

**Figure 5.2.** Figure 5.2: [Left] An example of MB scenario, where the robot (gray disc) travels along the [PITH_FULL_IMAGE:figures/full_fig_p131_5_2.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p131_5.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p132_5.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p133_5.png]

**Figure 5.3.** Figure 5.3: An example input of StabilNet when attempting to place a cup right on top of [PITH_FULL_IMAGE:figures/full_fig_p135_5_3.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p137_5.png]

**Figure 5.4.** Figure 5.4: Examples of disc instances. [Left] EE scenario with [PITH_FULL_IMAGE:figures/full_fig_p138_5_4.png]

**Figure 5.5.** Figure 5.5: Examples of instances with general-shaped objects in [Left] EE and [Right] [PITH_FULL_IMAGE:figures/full_fig_p138_5_5.png]

**Figure 5.6.** Figure 5.6: Algorithm performance in EE disc instances with [PITH_FULL_IMAGE:figures/full_fig_p139_5_6.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p139_5.png]

**Figure 5.7.** Figure 5.7: Algorithm performance in EE disc instances with [PITH_FULL_IMAGE:figures/full_fig_p140_5_7.png]

**Figure 5.8.** Figure 5.8: Algorithm performance in MB disc instances with [PITH_FULL_IMAGE:figures/full_fig_p140_5_8.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p141_5.png]

**Figure 5.9.** Figure 5.9: (a) A synthesized environment height map and the depth image of the placed object from the bottom. All objects are from the training set. (b) The corresponding stability prediction distribution of (a). (c) A synthesized environment height map and the depth image of …

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p142_5.png]

**Figure 5.10.** Figure 5.10: Algorithm performance in [Top] EE scenario and [Bottom] MB scenario with [PITH_FULL_IMAGE:figures/full_fig_p142_5_10.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p150_2.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p154_4.png]

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