Amazons Puzzles are NPComplete
• 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.
• 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 NPcomplete
• 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
• 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.
• 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
• Corollary: SAE = {p | Black wins simple
Amazons endgame p} is NP-equivalent (?)