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Amazons Puzzles are NP-Complete

Amazons Puzzles are NP-Complete. G ∞ is the infinite grid. Cubic Subgrid Graphs are subgraphs of G ∞ where nodes have degree at most three. HC3G = {G | G is a cubic subgrid graph with a Hamilton circuit} HCB3P = {G | G is a bipartite cubic planar graph with a Hamilton circuit}

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Amazons Puzzles are NP-Complete

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  1. Amazons Puzzles are NP-Complete

  2. G∞ is the infinite grid. • Cubic Subgrid Graphs are subgraphs of G∞ where nodes have degree at most three. • HC3G = {G | G is a cubic subgrid graph with a Hamilton circuit} • HCB3P = {G | G is a bipartite cubic planar graph with a Hamilton circuit} • HCB3P is known to be NP-complete, so we prove that HC3G is NP-complete by reducing from HCB3P to HC3G.

  3. Definition: A collision path is an edge disjoint path with at most one node repetition which ends right after the repetition • Corollary: The set of all cubic subgrid graphs G with a collision path of length |VG|-1 with a specified starting point is NP-complete

  4. Theorem: The set AP = {(p, b) | Amazons puzzle p has a solution length at least b} is NP-complete. • Proof by reduction from cubic subgrid graphs with a collision path. • Prove the equivalent statement: G has a collision path of length n-l (n=|VG|) starting in a specified node s if and only if the amazon can make at least b moves in position p

  5. Let m be the maximum number of moves the amazon can make in position p and L be the maximum length of collision paths in G. Let k = 6n (i.e., the corridors have length 12n). Set b to 12(n2 –n) to get: • Claim: L >= n-1  m >= 12(n2 –n) • m >= L(2k + 1) • m <= L(C + 2k + 1) +C, where C is the maximum number of empty squares in the 7x7 regions.

  6. Let C=11, k = 6n • Then m >= L(12n+1) and m <= (12n+12)+11 • Therefore: L >= n-1 => m >= (n-1)(12n+1) = 12n2-11n-1 L <= n-1 => m <= (n-2)(12n+12)+11 = 12n2-12n-13 • L >= n-1  m >= 12(n2-n) follows from this, and so AP is NP-complete

  7. Corollary: SAE = {p | Black wins simple Amazons endgame p} is NP-equivalent (?)

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