Mixed Guidance Graph Optimization (MGGO)

• MGGO is a guidance graph optimization method that combines directional constraints with action cost weighting to streamline lifelong multi-agent path finding.
• It employs a mixed regime where soft guidance via weights and strict directional guidance work together to prevent harmful head-on conflicts.
• MGGO leverages advanced optimization techniques, including evolutionary algorithms and quality diversity methods, to improve throughput across varied maps and planners.

Mixed Guidance Graph Optimization (MGGO) is a formulation for lifelong Multi-Agent Path Finding (LMAPF) in which guidance is represented not only by action costs but also by optimized edge directions. In this setting, a guidance graph does more than softly bias agents away from congested motions through higher weights: it can also impose strict directional structure by making selected traversals unidirectional. MGGO was introduced as a generalization of Guidance Graph Optimization (GGO), motivated by the observation that edge weights alone only discourage undesirable motions, whereas direction optimization can prohibit them and thereby reduce head-on conflicts under the rotational motion model used in lifelong MAPF (Zhang et al., 26 Feb 2026).

1. Historical position within guidance-graph methods

The immediate precursor to MGGO is GGO for LMAPF, where a bidirected weighted guidance graph is optimized so that agents plan on action costs rather than uniform grid costs. In that formulation, guidance is entirely weight-based: all legal movement directions remain available, and optimization acts only on their costs. The online continuation of this line of work further replaced static weights with a policy that updates guidance from real-time traffic patterns, but it still remained a weight-only mechanism (Zang et al., 2024).

MGGO changes the object being optimized. Instead of assuming that every traversable undirected edge should remain bidirected, it allows some non-bridge edges to become unidirectional while retaining optimized positive weights on all allowed actions. This creates a mixed regime of soft and strict guidance. Soft guidance is implemented by weights; strict guidance is implemented by edge directions. The resulting mixed guidance graph can therefore encode graded preferences and hard prohibitions simultaneously (Zhang et al., 26 Feb 2026).

This distinction is central to the method’s rationale. In prior GGO, an edge with a high weight only discourages use; it does not prevent use. The MGGO paper explicitly argues that this is insufficient when head-on conflicts are structurally harmful, especially under the rotational motion model, where detours may require additional rotations and can be more expensive than in the pebble model. A plausible implication is that MGGO is best understood not as a replacement for GGO, but as a strict-guidance extension of it.

2. Formal model and optimization problem

The environment is a connected graph $G(V,E)$, specialized in the experiments to four-connected grid maps under the rotational motion model. Each agent state consists of a current vertex and an orientation in $\{\text{North, South, East, West}\}$. At each timestep an agent can move forward, rotate $90^\circ$, or wait. Collisions occur when two agents occupy the same vertex at the same time or traverse the same edge in opposite directions at the same timestep. In this setting, standard MAPF minimizes sum-of-cost, whereas LMAPF maximizes throughput, defined as the average number of goal vertices reached at each timestep (Zhang et al., 26 Feb 2026).

Prior GGO uses a guidance graph

$G_g(V,E_g,\omegaV),$

with

$E_g = E_{wr} \cup E_{move},$

where $E_{wr}$ contains self-loops $(u,u)$ for waiting or rotating, and $E_{move}$ contains directed movement edges $(u,v)$, $u \neq v$. The edge weights satisfy

$\{\text{North, South, East, West}\}$0

GGO optimizes

$\{\text{North, South, East, West}\}$1

where $\{\text{North, South, East, West}\}$2 is the throughput objective.

MGGO replaces this with a mixed guidance graph

$\{\text{North, South, East, West}\}$3

defined so that for each non-bridge undirected edge $\{\text{North, South, East, West}\}$4, the optimizer may include $\{\text{North, South, East, West}\}$5, or $\{\text{North, South, East, West}\}$6, or both $\{\text{North, South, East, West}\}$7 and $\{\text{North, South, East, West}\}$8 in $\{\text{North, South, East, West}\}$9. For each bridge $90^\circ$0, both $90^\circ$1 and $90^\circ$2 must be present. For every $90^\circ$3, the self-loop $90^\circ$4 is also added. The edge weights satisfy

$90^\circ$5

MGGO is then posed as

$90^\circ$6

subject to

$90^\circ$7

and

$90^\circ$8

The strong-connectivity requirement is the key structural constraint. It ensures that every vertex remains reachable from every other vertex despite directional pruning. The bridge condition is a local consequence of that requirement: bridges must remain bidirected, because removing one bridge direction can destroy strong connectivity. This formalism makes the difference between GGO and MGGO precise. GGO optimizes only $90^\circ$9 on a bidirected action graph; MGGO optimizes both $G_g(V,E_g,\omegaV),$0 and $G_g(V,E_g,\omegaV),$1, turning the problem into a mixed discrete-continuous search over directed structure and continuous action costs (Zhang et al., 26 Feb 2026).

3. Algorithmic realizations

MGGO was introduced with two optimization methods and one related edge-direction-aware GGO extension. All three depend on fast simulation-based evaluation, and both MGGO methods rely on a repair primitive called Edge Reversal Search (ERS) to restore strong connectivity when candidate direction assignments violate it (Zhang et al., 26 Feb 2026).

ERS takes a mixed guidance graph $G_g(V,E_g,\omegaV),$2 and the bridge set $G_g(V,E_g,\omegaV),$3. It first restores bidirectionality on all bridges, then computes strongly connected components with Tarjan’s algorithm. If more than one SCC remains, it forms the condensation DAG, chooses a source SCC of indegree zero, collects outgoing edges from that SCC, reverses half of them, and repeats until only one SCC remains. Each ERS iteration costs

$G_g(V,E_g,\omegaV),$4

The procedure is heuristic rather than complete, but the paper reports that it always returned a valid solution in the tested instances.

The two MGGO methods differ in whether directions and weights are optimized sequentially or jointly.

Method	Optimized object	Main search method
Two-Phase MGGO-DS	Directions, then weights	$G_g(V,E_g,\omegaV),$5 EA, then CMA-ES
QD-based Joint MGGO-PU	CNN generating directions and weights	CMA-MAE
Edge-direction-aware GGO-PU	Weights only, with directional traffic input	CMA-ES

Two-Phase MGGO-DS decomposes the problem. In phase 1, each non-bridge edge $G_g(V,E_g,\omegaV),$6 receives a binary orientation choice: $G_g(V,E_g,\omegaV),$7 or $G_g(V,E_g,\omegaV),$8. The paper explicitly excludes a bidirected option at this stage because “an unweighted bidirected edge does not provide any guidance and, therefore, would be abandoned by the optimizer.” Candidate direction assignments are evaluated after ERS repair using a $G_g(V,E_g,\omegaV),$9 evolutionary algorithm, with throughput estimated from $E_g = E_{wr} \cup E_{move},$0 LMAPF simulations of horizon $E_g = E_{wr} \cup E_{move},$1. Initialization mixes three families: directed Crisscross, DFS-based strongly connected graphs, and random directed graphs. Phase 2 then fixes the resulting edge set $E_g = E_{wr} \cup E_{move},$2 and runs GGO-DS with CMA-ES to optimize $E_g = E_{wr} \cup E_{move},$3 (Zhang et al., 26 Feb 2026).

QD-based Joint MGGO-PU instead optimizes a parameterized update model that outputs both edge directions and edge weights. The search algorithm is Covariance Matrix Adaptation MAP-Annealing (CMA-MAE), a Quality Diversity method. The genotype is a CNN parameter vector; each sampled parameter vector instantiates a network that consumes precomputed traffic patterns and initial graph encodings and produces a mixed guidance graph. Two traffic regimes are precomputed by running $E_g = E_{wr} \cup E_{move},$4 LMAPF simulations on an unweighted guidance graph and on a directed Crisscross mixed guidance graph. Each simulation yields a tensor of shape $E_g = E_{wr} \cup E_{move},$5 encoding frequencies of four move directions, wait, clockwise rotation, and counterclockwise rotation. The weight channels use the established $E_g = E_{wr} \cup E_{move},$6 representation, while edge directions use a $E_g = E_{wr} \cup E_{move},$7 independent input encoding and a $E_g = E_{wr} \cup E_{move},$8 dependent output encoding so that local east-edge and south-edge relations can be one-hot over forward, reverse, or bidirected states. With $E_g = E_{wr} \cup E_{move},$9, the actual CNN input has size $E_{wr}$0 and the output $E_{wr}$1. The implemented network has 3 convolutional layers, kernel sizes $E_{wr}$2, $E_{wr}$3, $E_{wr}$4, 8 hidden channels per layer, ReLU and batch normalization, and 3,479 parameters (Zhang et al., 26 Feb 2026).

Joint MGGO-PU evaluates candidates with two objectives. The result objective $E_{wr}$5 is average throughput over $E_{wr}$6 simulations. The optimization objective is

$E_{wr}$7

where $E_{wr}$8 measures ERS-related direction similarity and $E_{wr}$9 in the reported experiments. Diversity is indexed by the ratio of unidirectional edges, which makes the archive explicitly explore the trade-off between stronger one-way traffic structure and greater bidirectional flexibility (Zhang et al., 26 Feb 2026).

Edge-direction-aware GGO-PU is not MGGO proper. It still optimizes only edge weights, but it augments the weight-update model with traffic patterns gathered from directed Crisscross guidance. The method therefore becomes direction-aware without outputting directed edges. The paper treats this distinction carefully because, empirically, direction-aware weight optimization can outperform full direction optimization in some regimes (Zhang et al., 26 Feb 2026).

4. Empirical behavior across planners and maps

The experiments evaluate eight setups across two LMAPF back ends: RHCR and PIBT. RHCR is tested on random-32-32-20 with $(u,u)$0, warehouse-33-36 with $(u,u)$1, and empty-48-48 with $(u,u)$2. PIBT is tested on random-32-32-20 with $(u,u)$3, warehouse-33-36 with $(u,u)$4, empty-48-48 with $(u,u)$5, room-64-64-8 with $(u,u)$6, and den312d with $(u,u)$7. Optimization budgets are standardized at

$(u,u)$8

with two-phase MGGO-DS splitting this into 5,000 evaluations for phase 1 and 15,000 for phase 2. Final reporting uses 50 simulation runs per optimized graph (Zhang et al., 26 Feb 2026).

A consistent finding is that one of the proposed methods is always best across the PIBT setups, but the winner varies by map. On warehouse-33-36 with PIBT, QD + Joint MGGO-PU reaches $(u,u)$9, compared with $E_{move}$0 for Two-Phase MGGO-DS, $E_{move}$1 for edge-direction-aware GGO-PU, and $E_{move}$2 for GGO-DS. On den312d with PIBT, Two-Phase MGGO-DS reaches $E_{move}$3, compared with $E_{move}$4 for QD + Joint MGGO-PU, $E_{move}$5 for edge-direction-aware GGO-PU, and $E_{move}$6 for GGO-DS. On empty-48-48 with PIBT, however, edge-direction-aware GGO-PU is best at $E_{move}$7, ahead of QD + Joint MGGO-PU at $E_{move}$8, Two-Phase MGGO-DS at $E_{move}$9, and GGO-DS at $(u,v)$0 (Zhang et al., 26 Feb 2026).

The RHCR results are more mixed and make the planner dependence explicit. On warehouse-33-36 with RHCR, QD + Joint MGGO-PU reaches $(u,v)$1, ahead of edge-direction-aware GGO-PU at $(u,v)$2, Two-Phase MGGO-DS at $(u,v)$3, and GGO-DS at $(u,v)$4. On empty-48-48 with RHCR, edge-direction-aware GGO-PU is best at $(u,v)$5, marginally above QD + Joint MGGO-PU at $(u,v)$6, with Two-Phase MGGO-DS at $(u,v)$7 and GGO-DS at $(u,v)$8. This pattern supports the paper’s interpretation that strict direction optimization helps less consistently for RHCR than for PIBT (Zhang et al., 26 Feb 2026).

The ablations illuminate why. For Joint MGGO-PU on setup 5, QD + Joint MGGO-PU achieves $(u,v)$9, whereas CMA-ES + Joint MGGO-PU achieves $u \neq v$0, and a version with independent edge-direction output representation achieves $u \neq v$1. The appendix further reports that QD found graphs with about 70% directed edges, while CMA-ES found only 14%, and that QD also reduced rotation ratio. For the phase-one EA in Two-Phase MGGO-DS, the phase-1 optimizer itself reaches $u \neq v$2; adding a bidirected option reduces this to $u \neq v$3, while restricting mutation diversity lowers it further to $u \neq v$4 or $u \neq v$5, depending on the operator set (Zhang et al., 26 Feb 2026).

These empirical patterns support two specific claims made by the paper. First, optimizing directions is especially valuable for PIBT, a locally reactive rule-based planner. Second, the benefit of direction optimization is map dependent: in open layouts such as empty-48-48, edge-direction-aware weight optimization can be stronger than explicit direction search.

5. Interpretation, trade-offs, and common misconceptions

A common misconception is that MGGO is simply “GGO plus more constraints.” In the LMAPF literature, the substantive difference is that MGGO introduces a second control channel. Weights shape preferences; directions shape reachability of individual actions. The paper’s formulation therefore changes both the search space and the operational semantics of guidance (Zhang et al., 26 Feb 2026).

A second misconception is that more one-way structure is necessarily better. The reported results do not support that. Two-Phase MGGO-DS tends to maximize the number of unidirectional edges because its phase-one search excludes bidirected choices for non-bridge edges. This can reduce the rotation action ratio, which is particularly valuable for PIBT under the rotational motion model, but it can also force longer detours and reduce flexibility. Joint MGGO-PU was introduced largely to expose this trade-off rather than to eliminate it: its Quality Diversity archive indexes solutions by the ratio of unidirectional edges precisely because different maps and planners favor different balances (Zhang et al., 26 Feb 2026).

A third misconception is that direction optimization uniformly dominates weight-only guidance. The paper explicitly shows otherwise. MGGO can outperform GGO, but not always, and edge-direction-aware GGO-PU can outperform full MGGO in some open-map regimes. This suggests that the marginal value of strict guidance depends on how much the underlying planner already exploits soft costs. The paper’s interpretation is that RHCR, being a stronger planner, can often avoid high-cost edges without needing hard directional prohibitions, whereas PIBT benefits more from having problematic local actions removed outright (Zhang et al., 26 Feb 2026).

A fourth misconception is that edge-direction-aware GGO-PU should be classified as MGGO. It should not. The method remains weight-only and does not alter graph topology. Its significance is methodological: it shows that directional traffic statistics are useful even when the optimized object remains a bidirected graph.

6. Limitations and research directions

The paper identifies reduced sample efficiency as a primary limitation. MGGO enlarges the search space from continuous edge weights to a mixed discrete-continuous object, while all evaluations remain simulation based. This makes optimization harder even when final throughput improves. The authors explicitly suggest surrogate modeling as future work for improving sample efficiency (Zhang et al., 26 Feb 2026).

A second limitation is that ERS is heuristic. It is fast, with $u \neq v$6 per iteration, and empirically successful in the reported experiments, but it is not presented as a complete or optimal repair algorithm. The repair-related regularizer in Joint MGGO-PU further implies that generation and repair are not yet perfectly aligned.

A third limitation is representational. The experiments are centered on grid maps and the rotational motion model, and the traffic encodings used by Joint MGGO-PU are explicitly tensorized as $u \neq v$7. This suggests that the current implementations are specialized to lattice environments. A plausible implication is that extending MGGO to arbitrary graphs would require a different graph generator and different traffic descriptors.

A fourth limitation is that the reported MGGO methods are offline. In parallel work on online guidance graph optimization, a policy updates edge weights from real-time traffic and changing task distributions to improve PIBT and GPIBT (Zang et al., 2024). That paper does not optimize directions, but it demonstrates that adaptive weight guidance can outperform both static guidance and human-designed online policies under non-stationary task distributions. This suggests a direct next step: an online MGGO in which edge directions and weights are jointly adapted from real-time traffic rather than optimized only once offline.

In that sense, MGGO currently occupies a specific place in the guidance-graph literature. It is the strict-guidance generalization of offline GGO for LMAPF, and it exposes a design space in which one-way traffic structure, bidirectional flexibility, and continuous cost shaping are all optimization variables rather than hand-designed conventions.