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Graph-Based Heuristics for Minimum Constraint Removal in Cluttered Robot Motion Planning

Antony Thomas, Fulvio Mastrogiovanni, Marco Baglietto

arXiv:2610.12609v1cs.RO

TL;DR

When clutter makes robot motion infeasible if all objects are treated as fixed, MCR seeks a minimum set of obstacles to remove. The paper combines bidirectional feasible-set generation with reduced-graph refinement and an extended generator for candidate-coverage cases. Across six benchmarks, the pipeline matches exact solutions in three cases and is faster than exact search, with a worst observed solution-cardinality ratio of 1.31.

  • Problem

    MCR seeks to identify the minimum number of movable obstacles whose removal restores a collision-free path, but computing an optimal solution is NP-hard.

    Source: §I · p. 1

  • Method

    The paper combines bidirectional feasible-set generation with fixed-length searches on reduced MCR graphs, extending candidate generation when GFS-1 candidates may not contain a complete optimal set.

    Source: §I · pp. 1–2

  • Results

    Across six benchmarks, GFS-1+Refine matched the exact solution in three cases, had a worst solution-cardinality ratio of 1.31, and ran 3.44–33.71 times faster than exact search.

    Source: §V · p. 7

  • Takeaways & Limitations

    In the manipulation benchmark, selecting two rather than seven obstacles reduced combined MCR, inverse-kinematics, and motion-planning time from 84.72 s to 30.59 s.

    Source: §V · p. 7

    GFS-1 candidates may not contain a complete optimal removal set, motivating the extended GFS-2 generator.

    Source: §I · p. 1

Abstract

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Humanoid and service robots operating in clut- tered human environments may encounter motion planning problems that are infeasible when all surrounding objects are treated as fixed obstacles. The Minimum Constraint Removal problem, which we refer to as MCR, addresses this situation by identifying the minimum number of constraints, such as movable objects, whose removal enables a collision-free path. However, computing an optimal MCR solution is NP-hard. In this work, we propose a graph-based heuristic pipeline that combines bidirectional feasible-set generation with fixed- length searches over reduced MCR graphs. We further derive an extended feasible-set generator for instances in which the candidates produced by the first generator do not contain a complete optimal removal set. Across six benchmarks, the evaluated pipeline matches the exact solution in three cases and, in the remaining cases, selects at most 31.25% more obstacles than the optimum, while being between 3.4 and 33.7 times faster than exact search. In one of the benchmarks, involving a manipulation task amidst clutter, selecting two rather than seven obstructing objects reduces the combined MCR, inverse- kinematics, and motion planning time from 84.72 s to 30.59 s. These results indicate that the proposed approach provides a practical compromise between computational efficiency and solution quality for motion planning among movable obstacles.

I. INTRODUCTION

In cluttered robot environments, MCR identifies a minimum set of movable obstacles to remove when treating all objects as fixed makes motion infeasible. The paper proposes feasible-set generation and reduced-graph refinement to seek low-cardinality solutions efficiently despite MCR’s computational difficulty.

I. INTRODUCTION

  • MCR formalizes the goal of finding the smallest set of objects whose relocation restores motion feasibility, avoiding additional perception, manipulation planning, and execution interactions where possible.MCR is NP-hard, and exact search can be expensive while aggressive greedy pruning can return solutions with substantially more obstacles.Source: §I · p. 1
  • The pipeline combines bidirectional feasible-set generation with refinement through fixed-length searches on reduced MCR graphs.GFS-1 uses shared obstacle labels at the two search frontiers to construct candidate removal sets.Source: §I · pp. 1–2
  • GFS-2 extends candidate generation for cases where GFS-1 candidates may not contain a complete optimal removal set.The paper also identifies computational-complexity analysis of this extended generator as a contribution.Source: §I · pp. 1–2
  • Across six benchmarks, GFS-1 + Refine matched the exact solution in three cases, selected at most 31.25% more obstacles than optimal otherwise, and achieved 3.4–33.7× speedups over exact search.In the manipulation benchmark, it selected two objects rather than seven, reducing combined MCR, inverse-kinematics, and motion-planning time from 84.72 s to 30.59 s.Source: §I · p. 2
  • In the manipulation benchmark, selecting two rather than seven obstacles reduced combined MCR, inverse-kinematics, and motion-planning time from 84.72 s to 30.59 s.The comparison is against greedy search.Source: §I · p. 2

II. PRELIMINARIES

The preliminaries formulate MCR as finding a minimum-cardinality obstacle set whose removal admits a collision-free path, then represent the problem as graph search. Because repeated encounters with an obstacle count only once, locally shortest paths need not combine into a globally optimal path.

II. PRELIMINARIES

  • Removing an obstacle abstracts relocating its object outside the region relevant to the planned motion; MCR assigns unit removal cost and does not plan relocation.The formulation does not account for whether one object is harder to manipulate than another.Source: §II · pp. 2–3
  • MCR seeks a minimum-cardinality set of obstacles whose removal admits a path from start to goal that avoids all remaining obstacles.If no removal is needed, the optimal set is empty; any one of several minimum sets may be used.Source: §II · p. 2
  • The graph represents connected configuration-space cells as vertices, with edges between adjacent cells and labels listing the obstacles intersecting each cell.A free-space cell has an empty obstacle label.Source: §II · p. 2, Figure 2 · p. 3
  • A path’s removable set is a union, so an obstacle encountered multiple times contributes only once to its cost.The partial-path notation tracks removable sets along paths and path segments.Source: §II · p. 2
  • Exact best-first search orders states by accumulated removal-set size for completeness and optimality, but must retain multiple sets at a vertex because their future costs can differ.Keeping only the smallest set at each vertex can discard a path that later yields a better global solution.Source: §II · p. 3
  • At a vertex labeled with {o3}, combining locally minimum subpaths can yield {o1, o2, o3, o4}, although an s–g path through that vertex uses only {o1, o2, o3}.Thus a globally optimal path may have a prefix that is not locally optimal by removal-set cardinality.Source: §II · p. 3

A. Rationale

The rationale is to generate feasible obstacle-removal candidates, then refine them through bounded-length searches on reduced MCR graphs.

A. Rationale

  • A candidate is feasible when its induced graph connects s and g, corresponding to a collision-free path after removing its obstacles.The induced graph uses vertices whose obstacle labels are subsets of the candidate set.Source: A. Rationale · p. 3
  • GFS-1 explores the MCR graph bidirectionally from the start and goal, expanding the respective frontiers.It operates on a working graph whose obstacle labels can change while the original labels remain unchanged.Source: A. Rationale · p. 3
  • At each depth, labels occurring in both frontiers are added to the current candidate set.The generator collects obstacle labels represented in the two frontiers before identifying their overlap.Source: A. Rationale · p. 3
  • Removing shared labels and contracting vertices whose labels become empty preserves free-space connectivity; previously generated labels are excluded from subsequent candidates.The resulting nonempty GFS-1 candidates are disjoint, and their number is at most the number of obstacles.Source: A. Rationale · pp. 3–4

B. Reduced-Graph Candidate Refinement

Reduced-graph refinement independently searches feasible candidates using fixed path lengths, while GFS-2 extends candidate generation when GFS-1 may omit an optimal set.

B. Reduced-Graph Candidate Refinement

  • Refine examines each feasible candidate independently because the smallest generated candidate need not contain the best refined solution.A GFS-1 candidate is feasible but need not be minimal.Source: B. Reduced-Graph Candidate Refinement · p. 4
  • For each candidate, Refine searches selected path lengths in its reduced graph and returns the path’s minimum-cardinality removable set.Every s–g path in the reduced graph has a removable set contained in the candidate.Source: B. Reduced-Graph Candidate Refinement · p. 4
  • The admissible lengths run from 2l through 2l + r_C − 1, with additional lengths when frontiers are adjacent to the opposite terminal.Path length is measured in graph edges, and r_C is the maximum multiplicity of an obstacle label in the reduced graph.Source: B. Reduced-Graph Candidate Refinement · p. 4
  • At the same vertex and depth, a state can be pruned when another state’s removable set is a subset; cardinality alone is not a valid dominance test.Source: B. Reduced-Graph Candidate Refinement · p. 4
  • Although |C2| < |C1|, the optimal set {o5, o6} lies in C1, whose reduced-graph search can recover a solution smaller than either candidate.Source: B. Reduced-Graph Candidate Refinement · p. 4
  • Candidate containment is insufficient for recovering an optimum because its optimal path may have a length outside the searched lengths, so the pipeline has no worst-case approximation guarantee.Source: B. Reduced-Graph Candidate Refinement · p. 4

C. Extended Feasible-Set Generation

GFS-2 addresses cases where GFS-1 candidates contain no complete optimal removal set. It continues the bidirectional search across increasing depths so that refinement can use larger candidates.

C. Extended Feasible-Set Generation

  • GFS-1 may generate candidates that contain no complete optimal removal set, preventing refinement from recovering the optimum.Recovery also requires searching the associated optimal path length.Source: C. Extended Feasible-Set Generation · p. 5
  • GFS-2 retains labels, frontiers, and visited sets across restarts, allowing it to combine labels that GFS-1 separates.It continues at increasing depths and can generate larger candidates.Source: C. Extended Feasible-Set Generation · p. 5
  • In the Figure 5 instance, GFS-2 generates a candidate containing the optimum {o1, o4}, which refinement recovers using a path of length three.GFS-1's candidate in this instance is {o1, o2, o3}, which does not contain the optimum.Source: C. Extended Feasible-Set Generation · p. 5

D. Complexity Analysis

The analysis bounds traversal, contraction, candidate generation, and refinement using MCR graph size and degree. It concludes that the pipeline remains exponential in the worst case despite practical benefits from dominance pruning.

D. Complexity Analysis

  • A bidirectional pass costs O(N + M) for traversal and O(N∆^2) for contracting at most N vertices.The bounds assume bit-set obstacle labels and an inverted index from labels to vertices.Source: D. Complexity Analysis · p. 5
  • GFS-1 generates at most n nonempty candidates because previously used labels are excluded and its candidates are disjoint.Here q1 is the number of GFS-1 candidates and n is the number of obstacles.Source: D. Complexity Analysis · p. 5
  • GFS-2 performs at most N bidirectional depth expansions; q2 feasible candidates may require up to q2 separate contraction sequences.Its frontiers and visited sets are not reset between expansions.Source: D. Complexity Analysis · p. 5
  • For each candidate, constructing the reduced graph costs O(N + M), while enumerating fixed-length paths has worst-case cost O(∆_C^L).∆_C is the candidate graph's maximum degree and L is the searched path length.Source: D. Complexity Analysis · p. 5
  • The full pipeline is a heuristic, not claimed to run in polynomial time, because its worst-case dependence on maximum searched path length remains exponential.Subset-based dominance pruning reduces explored states in practice but does not change that dependence.Source: D. Complexity Analysis · pp. 5–6

IV. EVALUATION

Across six benchmarks, GFS-1+Refine is compared with exact and greedy search for solution quality and computation time. It matches the optimum in three cases and is faster than exact search, while a manipulation benchmark shows lower combined planning time with a smaller removal set.

IV. EVALUATION

  • The baselines were exact best-first MCR search and greedy search, and all methods used the same precomputed MCR graph.Reported graph-search times exclude graph construction and average 10 repeated runs.Source: §IV · p. 6, Table I · p. 7
  • All six benchmarks satisfy GFS-1's candidate-coverage condition, so GFS-2 was not invoked in the reported experiments.The evaluated pipeline was therefore GFS-1+Refine.Source: §IV · p. 6
  • Across six benchmarks, the method matches the optimum in three, with worst observed ratio 1.31 and mean ratio 1.11; it runs 3.44 to 33.71 times faster than exact search.Table I reports graph-search results for the six benchmarks.Source: §IV · pp. 6–7, Table I · p. 7
  • 30.59 s versus 84.72 s was the combined planning time for GFS-1+Refine's two selected objects and greedy search's seven objects in S6.The combined time includes MCR search, inverse kinematics, and motion planning.Source: Figure 7 · p. 6, §IV · p. 7
  • Although greedy search is faster in isolation, its larger removal set can increase subsequent manipulation-planning effort.The S6 experiment measures computational planning cost, not physical execution time.Source: §IV · p. 7

V. CONCLUSIONS

The paper presents GFS-1+Refine for feasible, low-cardinality MCR solutions and GFS-2 for cases where GFS-1 candidates omit a complete optimal removal set. Across six benchmarks, GFS-1+Refine matched exact search in three, with a worst observed solution-cardinality ratio of 1.31 and speed-ups of 3.44–33.71.

V. CONCLUSIONS

  • GFS-1+Refine generates feasible, low-cardinality MCR solutions, while GFS-2 extends feasible-set generation when GFS-1 candidates do not contain a complete optimal removal set.The passage describes the pipeline as a heuristic approach.Source: §V · p. 7
  • GFS-1+Refine matched the exact solution in three of six benchmarks, with a worst observed solution-cardinality ratio of 1.31 and speed-ups from 3.44 to 33.71.Its median speed-up over exact search was 5.72.Source: §V · p. 7
  • In the cluttered manipulation benchmark, selecting two rather than seven objects reduced combined MCR, inverse-kinematics, and motion-planning time from 84.72 s to 30.59 s.The paper reports that additional computation during obstacle selection can reduce subsequent manipulation-planning effort.Source: §V · p. 7
  • The approach has no worst-case approximation guarantee, assigns equal cost to every object removal, and does not compute or execute relocation motions.The paper identifies object-dependent manipulation costs, grasp and placement feasibility, and humanoid whole-body validation as future work.Source: §V · p. 7
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