Map-Based Constraint Optimization Strategy
A systematic approach to solving placement optimization problems on spatial maps. This applies to any problem where you must place items on a grid to maximize an objective while respecting placement constraints.
Why Exhaustive Search Fails
Exhaustive search (brute-force enumeration of all possible placements) is the worst approach:
- Combinatorial explosion: Placing N items on M valid tiles = O(M^N) combinations
- Even small maps become intractable (e.g., 50 tiles, 5 items = 312 million combinations)
- Most combinations are clearly suboptimal or invalid
The Three-Phase Strategy
Phase 1: Prune the Search Space
Goal: Eliminate tiles that cannot contribute to a good solution.
Remove tiles that are:
- Invalid for any placement - Violate hard constraints (wrong terrain, out of range, blocked)
- Dominated - Another tile is strictly better in all respects
- Isolated - Too far from other valid tiles to form useful clusters
Before: 100 tiles in consideration
After pruning: 20-30 candidate tiles
This alone can reduce search space by 70-90%.
Phase 2: Identify High-Value Spots
Goal: Find tiles that offer exceptional value for your objective.
Score each remaining tile by:
- Intrinsic value - What does this tile contribute on its own?
- Adjacency potential - What bonuses from neighboring tiles?
- Cluster potential - Can this tile anchor a high-value group?
Rank tiles and identify the top candidates. These are your priority tiles - any good solution likely includes several of them.
Example scoring:
- Tile A: +4 base, +3 adjacency potential = 7 points (HIGH)
- Tile B: +1 base, +1 adjacency potential = 2 points (LOW)
Phase 3: Anchor Point Search
Goal: Find placements that capture as many high-value spots as possible.
- Select anchor candidates - Tiles that enable access to multiple high-value spots
- Expand from anchors - Greedily add placements that maximize marginal value
- Validate constraints - Ensure all placements satisfy requirements
- Local search - Try swapping/moving placements to improve the solution
For problems with a "center" constraint (e.g., all placements within range of a central point):
- The anchor IS the center - try different center positions
- For each center, the reachable high-value tiles are fixed
- Optimize placement within each center's reach
Algorithm Skeleton
def optimize_placements(map_tiles, constraints, num_placements):
# Phase 1: Prune
candidates = [t for t in map_tiles if is_valid_tile(t, constraints)]
# Phase 2: Score and rank
scored = [(tile, score_tile(tile, candidates)) for tile in candidates]
scored.sort(key=lambda x: -x[1]) # Descending by score
high_value = scored[:top_k]
# Phase 3: Anchor search
best_solution = None
best_score = 0
for anchor in get_anchor_candidates(high_value, constraints):
solution = greedy_expand(anchor, candidates, num_placements, constraints)
solution = local_search(solution, candidates, constraints)
if solution.score > best_score:
best_solution = solution
best_score = solution.score
return best_solution
Key Insights
Prune early, prune aggressively - Every tile removed saves exponential work later
High-value tiles cluster - Good placements tend to be near other good placements (adjacency bonuses compound)
Anchors constrain the search - Once you fix an anchor, many other decisions follow logically
Greedy + local search is often sufficient - You don't need the global optimum; a good local optimum found quickly beats a perfect solution found slowly
Constraint propagation - When you place one item, update what's valid for remaining items immediately
Common Pitfalls
- Ignoring interactions - Placing item A may change the value of placing item B (adjacency effects, mutual exclusion)
- Over-optimizing one metric - Balance intrinsic value with flexibility for remaining placements
- Forgetting to validate - Always verify final solution satisfies ALL constraints
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