This is a minimum vertex cover problem on a bipartite graph, which I did not see at first.
Model the problem as a minimum vertex cover on a bipartite graph where left nodes are rows and right nodes are columns, and each hole at (i, j) creates an edge between row i and column j. By König's theorem, the minimum vertex cover size equals the maximum matching size, which can be computed efficiently using Hopcroft-Karp or DFS-based augmenting paths.
Pro tip: Clarify that boards can overlap and cover intact cells, so covering a hole only requires selecting its row or column; this reduces to covering all edges with minimum vertices. Mention that the problem is equivalent to finding the minimum number of lines (rows or columns) to cover all 0s in a binary matrix, a classic problem solvable in polynomial time.
Restate the problem: given an m×n binary grid with 0s as holes, find the minimum number of full rows or full columns to cover all holes. Note that boards can overlap and cover intact cells, so covering a hole only requires selecting its row or column.
Create a bipartite graph with left nodes representing rows and right nodes representing columns. For each hole at (i, j), add an edge between row i and column j. The goal is to select a minimum set of vertices (rows/columns) that covers all edges, i.e., a minimum vertex cover.
Use König's theorem: in bipartite graphs, the size of the minimum vertex cover equals the size of the maximum matching. Thus, the problem reduces to finding a maximum matching in the bipartite graph.
Implement a maximum matching algorithm such as Hopcroft-Karp (O(E√V)) or a simpler DFS-based augmenting path algorithm (O(VE)). Since the graph is bipartite, these are efficient for typical grid sizes.
The size of the maximum matching is the minimum number of boards needed. Optionally, reconstruct the actual set of rows and columns from the matching to provide the specific boards.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.