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:

1. Invalid for any placement - Violate hard constraints (wrong terrain, out of range, blocked)
2. Dominated - Another tile is strictly better in all respects
3. Isolated - Too far from other valid tiles to form useful clusters

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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:

1. Intrinsic value - What does this tile contribute on its own?
2. Adjacency potential - What bonuses from neighboring tiles?
3. 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.

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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.

1. Select anchor candidates - Tiles that enable access to multiple high-value spots
2. Expand from anchors - Greedily add placements that maximize marginal value
3. Validate constraints - Ensure all placements satisfy requirements
4. 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#

python

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#

1. Prune early, prune aggressively - Every tile removed saves exponential work later

2. High-value tiles cluster - Good placements tend to be near other good placements (adjacency bonuses compound)

3. Anchors constrain the search - Once you fix an anchor, many other decisions follow logically

4. 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

5. 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