SKILL.md: DAG Chain Decomposition for Parallel Execution
When to Use This Skill
Load this skill when you encounter dependency networks that require optimal parallel execution with minimal coordination overhead. Key triggers:
- Task decomposition with ordering constraints but parallelizable components
- Multi-agent coordination where agent count impacts efficiency
- Resource allocation under dependencies
- Workflow optimization with inherent parallelism
DECISION POINTS
1. Decomposition Strategy Selection
IF problem has clear topological levels (stratifiable)
AND width ≤ √n
THEN use Chen's stratification algorithm
→ Stratify into levels V₁, V₂, ..., Vₕ
→ Apply maximum matching between adjacent levels
ELSE IF width >> √n (width-dominated problem)
THEN accept minimal compression possible
→ Create b chains (one per antichain element)
→ Focus on efficient chain management, not decomposition
ELSE IF dependencies are tangled (no clear levels)
THEN question dependency model validity
→ Look for missing abstractions
→ Consider alternative graph representations
2. Virtual Node vs. Immediate Assignment
IF task cannot be assigned to existing chains
AND remaining depth ≥ 3 levels
AND assignment would violate dependencies
THEN create virtual node for deferred resolution
→ Propagate virtual node to next level
→ Resolve in second phase with full context
ELSE IF remaining depth < 3 levels
THEN create new chain immediately
→ Overhead of deferred resolution exceeds benefit
3. Resource Allocation Strategy
WHEN assigning level Vᵢ to existing chains
FIRST formulate bipartite matching:
- Left: tasks in Vᵢ needing assignment
- Right: existing chains from previous levels
- Edges: assignments that preserve dependencies
IF maximum matching covers all tasks
THEN assign per matching (no new chains needed)
ELSE unmatched tasks each spawn new chain
→ This minimizes total chain count
→ Provably optimal by maximum matching theory
4. Width Measurement and Validation
TO measure problem width (coordination lower bound):
1. Identify all mutually incomparable elements (antichain)
2. Find maximum antichain size = width b
IF achieved chain count = b
THEN decomposition is optimal
ELSE IF achieved count > b
THEN investigate inefficiency
→ Check for premature commitment
→ Verify stratification correctness
→ Look for missed matching opportunities
FAILURE MODES
1. Sub-Width Compression Fallacy
Symptoms: Attempting to use fewer than b chains, or claiming decomposition into k < b chains
Detection: Count maximum antichain size; if solution uses fewer chains than antichain size, it's impossible
Fix: Measure width first, accept it as absolute lower bound, optimize other aspects
2. Premature Assignment Thrashing
Symptoms: Creating many short chains instead of fewer long ones; poor chain utilization
Detection: Chain count significantly exceeds width; many chains with only 1-2 tasks
Fix: Use virtual nodes for deferred decisions; resolve assignments only when full context available
3. Depth-Width Confusion
Symptoms: Treating long dependency chains as coordination problems; over-parallelizing sequential workflows
Detection: Trying to parallelize tasks that must run sequentially; creating false dependencies
Fix: Distinguish depth (latency) from width (coordination); long chains ≠ coordination complexity
4. Local Optimization Without Stratification
Symptoms: Greedy assignment without global structure; decisions that block future optimal assignments
Detection: Assignment quality degrades with problem size; no clear levelization strategy
Fix: Stratify first to isolate levels, then apply maximum matching within each level interface
5. Resource Spawning Before Matching
Symptoms: Creating new chains/agents without fully utilizing existing ones
Detection: Chain utilization is low; new resources created while existing ones could handle tasks
Fix: Apply maximum matching to existing resources before spawning new ones
WORKED EXAMPLES
Example 1: Multi-Agent Task Coordination
Scenario: Deploying software across 3 environments (dev, staging, prod) with 8 microservices, where some services depend on others being deployed first.
Dependencies:
- Database services must deploy before application services
- Shared libraries before services that use them
- Cross-environment promotion only after full environment validation
Step 1 - Width Analysis:
Level analysis reveals maximum antichain: {DB-dev, DB-staging, DB-prod, SharedLib-dev, SharedLib-staging, SharedLib-prod}
Width b = 6 (these can all run in parallel)
Minimum agents needed = 6
Step 2 - Stratification:
V₁: {DB-dev, DB-staging, DB-prod, SharedLib-dev, SharedLib-staging, SharedLib-prod} (6 nodes)
V₂: {App1-dev, App1-staging, App1-prod, App2-dev, App2-staging, App2-prod} (6 nodes)
V₃: {Integration-tests-dev, Integration-tests-staging, Integration-tests-prod} (3 nodes)
V₄: {Promotion-to-staging, Promotion-to-prod} (2 nodes)
Step 3 - Chain Assignment:
V₁ → V₂ matching:
- Chain 1: DB-dev → App1-dev
- Chain 2: DB-staging → App1-staging
- Chain 3: DB-prod → App1-prod
- Chain 4: SharedLib-dev → App2-dev
- Chain 5: SharedLib-staging → App2-staging
- Chain 6: SharedLib-prod → App2-prod
V₂ → V₃ matching produces 3 unmatched tasks (integration tests), but we can extend existing chains:
- Chain 1 extended: DB-dev → App1-dev → Integration-tests-dev
- Chain 2 extended: DB-staging → App1-staging → Integration-tests-staging
- Chain 3 extended: DB-prod → App1-prod → Integration-tests-prod
- Chains 4,5,6 complete after V₂
Result: 6 parallel agents (optimal), each handling one complete deployment pipeline.
Example 2: Parallel Data Pipeline with Deferred Decisions
Scenario: Processing sensor data where some transformations depend on data quality metrics that aren't known until runtime.
Initial structure: Raw data → Quality check → [Unknown branching] → Aggregation → Output
Challenge: Can't determine optimal assignment until quality metrics computed, but want to minimize coordination overhead.
Solution using virtual nodes:
Level 1: Raw data ingestion (parallel by sensor type)
Level 2: Quality assessment (one per data stream)
Level 3: Virtual nodes for "post-quality processing" (deferred)
Level 4: Aggregation and output
Maximum matching L1→L2: Direct assignment (each sensor to quality checker)
L2→L3: Create virtual nodes since branching strategy unknown
L3→L4: Resolve during execution based on actual quality metrics
Two-phase resolution:
- Structure phase: Establish pipeline skeleton with virtual placeholders
- Runtime phase: Resolve virtual nodes based on computed quality metrics
Benefit: Avoids premature commitment to processing strategy while ensuring resource utilization stays optimal.
QUALITY GATES
Decomposition is complete when ALL conditions are satisfied:
NOT-FOR Boundaries
DO NOT use this skill for:
- Independent parallel tasks: If width = n (no dependencies), use simple parallel dispatch instead → Use:
parallel-task-dispatch
- Purely sequential workflows: If width = 1, no decomposition benefit → Use:
sequential-pipeline-optimization
- Real-time scheduling: This optimizes structure, not timing → Use:
real-time-task-scheduling
- Dynamic dependencies: When dependency graph changes frequently → Use:
adaptive-workflow-management
- Resource-constrained allocation: When you have fewer resources than optimal chains → Use:
resource-constrained-scheduling
- Event-driven architectures: When tasks are triggered by external events → Use:
event-driven-orchestration
Delegate to other skills when:
- Problem has < 10 nodes → Use simple dependency resolution
- Dependencies are cyclic → Use:
cycle-breaking-strategies
- Need load balancing across resources → Use:
dynamic-load-balancing
- Optimizing for latency rather than throughput → Use:
critical-path-optimization