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"Minimum Falling Path Sum" (MFPS) is classic question in programming - "Givena grid of size $N{\times}N$ with integers in cells, return the minimum sum of afalling path through grid. A falling path starts at any cell in the first rowand ends in last row, with the rule of motion - the next element after the cell$(i,j)$ is one of the cells $(i+1,j-1),(i+1,j)$ and $(i+1,j+1)$". This problemhas linear solution (LS) (i.e. O($N^2$)) using dynamic programming method(DPM). There is an Multi-Agent version of MFPS called "Cherry Pickup II" (CP2).CP2 is a search for the maximum sum of 2 falling paths started from topcorners, where each covered cell summed up one time. All known fast solutionsof CP2 uses DPM, but have O($N^3$) time complexity on grid $N{\times}N$. Herewe offer a LS of CP2 (also using DPM) as finding maximum total weight of 2vertex-disjoint paths. Also, we extend this LS for some extended version of CP2with wider motion rules.

On Linear Solution of "Cherry Pickup II". Max Weight of Two Disjoint Paths in Node-Weighted Gridlike DAG