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/* A solver for the Japanese number-puzzle Hashiwokakero * (http://en.wikipedia.org/wiki/Hashiwokakero) * * Sebastian Nowozin <nowozin@gmail.com>, 13th January 2009 */
param n := 25;set rows := 1..n;set cols := 1..n;param givens{rows, cols}, integer, >= 0, <= 8, default 0;
/* Set of vertices as (row,col) coordinates */set V := { (i,j) in { rows, cols }: givens[i,j] != 0 };
/* Set of feasible horizontal edges from (i,j) to (k,l) rightwards */set Eh := { (i,j,k,l) in { V, V }: i = k and j < l and # Same row and left to right card({ (s,t) in V: s = i and t > j and t < l }) = 0 # No vertex inbetween };
/* Set of feasible vertical edges from (i,j) to (k,l) downwards */set Ev := { (i,j,k,l) in { V, V }: j = l and i < k and # Same column and top to bottom card({ (s,t) in V: t = j and s > i and s < k }) = 0 # No vertex inbetween };
set E := Eh union Ev;
/* Indicators: use edge once/twice */var xe1{E}, binary;var xe2{E}, binary;
/* Constraint: Do not use edge or do use once or do use twice */s.t. edge_sel{(i,j,k,l) in E}: xe1[i,j,k,l] + xe2[i,j,k,l] <= 1;
/* Constraint: There must be as many edges used as the node value */s.t. satisfy_vertex_demand{(s,t) in V}: sum{(i,j,k,l) in E: (i = s and j = t) or (k = s and l = t)} (xe1[i,j,k,l] + 2.0*xe2[i,j,k,l]) = givens[s,t];
/* Constraint: No crossings */s.t. no_crossing1{(i,j,k,l) in Eh, (s,t,u,v) in Ev: s < i and u > i and j < t and l > t}: xe1[i,j,k,l] + xe1[s,t,u,v] <= 1;s.t. no_crossing2{(i,j,k,l) in Eh, (s,t,u,v) in Ev: s < i and u > i and j < t and l > t}: xe1[i,j,k,l] + xe2[s,t,u,v] <= 1;s.t. no_crossing3{(i,j,k,l) in Eh, (s,t,u,v) in Ev: s < i and u > i and j < t and l > t}: xe2[i,j,k,l] + xe1[s,t,u,v] <= 1;s.t. no_crossing4{(i,j,k,l) in Eh, (s,t,u,v) in Ev: s < i and u > i and j < t and l > t}: xe2[i,j,k,l] + xe2[s,t,u,v] <= 1;
/* Model connectivity by auxiliary network flow problem: * One vertex becomes a target node and all other vertices send a unit flow * to it. The edge selection variables xe1/xe2 are VUB constraints and * therefore xe1/xe2 select the feasible graph for the max-flow problems. */set node_target := { (s,t) in V: card({ (i,j) in V: i < s or (i = s and j < t) }) = 0};set node_sources := { (s,t) in V: (s,t) not in node_target };
var flow_forward{ E }, >= 0;var flow_backward{ E }, >= 0;s.t. flow_conservation{ (s,t) in node_target, (p,q) in V }: /* All incoming flows */ - sum{(i,j,k,l) in E: k = p and l = q} flow_forward[i,j,k,l] - sum{(i,j,k,l) in E: i = p and j = q} flow_backward[i,j,k,l] /* All outgoing flows */ + sum{(i,j,k,l) in E: k = p and l = q} flow_backward[i,j,k,l] + sum{(i,j,k,l) in E: i = p and j = q} flow_forward[i,j,k,l] = 0 + (if (p = s and q = t) then card(node_sources) else -1);
/* Variable-Upper-Bound (VUB) constraints: xe1/xe2 bound the flows. */s.t. connectivity_vub1{(i,j,k,l) in E}: flow_forward[i,j,k,l] <= card(node_sources)*(xe1[i,j,k,l] + xe2[i,j,k,l]);s.t. connectivity_vub2{(i,j,k,l) in E}: flow_backward[i,j,k,l] <= card(node_sources)*(xe1[i,j,k,l] + xe2[i,j,k,l]);
/* A feasible solution is enough */minimize cost: 0;
solve;
/* Output solution graphically */printf "\nSolution:\n";for { row in rows } { for { col in cols } { /* First print this cell information: givens or space */ printf{0..0: givens[row,col] != 0} "%d", givens[row,col]; printf{0..0: givens[row,col] = 0 and card({(i,j,k,l) in Eh: i = row and col >= j and col < l and xe1[i,j,k,l] = 1}) = 1} "-"; printf{0..0: givens[row,col] = 0 and card({(i,j,k,l) in Eh: i = row and col >= j and col < l and xe2[i,j,k,l] = 1}) = 1} "="; printf{0..0: givens[row,col] = 0 and card({(i,j,k,l) in Ev: j = col and row >= i and row < k and xe1[i,j,k,l] = 1}) = 1} "|"; printf{0..0: givens[row,col] = 0 and card({(i,j,k,l) in Ev: j = col and row >= i and row < k and xe2[i,j,k,l] = 1}) = 1} '"'; printf{0..0: givens[row,col] = 0 and card({(i,j,k,l) in Eh: i = row and col >= j and col < l and (xe1[i,j,k,l] = 1 or xe2[i,j,k,l] = 1)}) = 0 and card({(i,j,k,l) in Ev: j = col and row >= i and row < k and (xe1[i,j,k,l] = 1 or xe2[i,j,k,l] = 1)}) = 0} " ";
/* Now print any edges */ printf{(i,j,k,l) in Eh: i = row and col >= j and col < l and xe1[i,j,k,l] = 1} "-"; printf{(i,j,k,l) in Eh: i = row and col >= j and col < l and xe2[i,j,k,l] = 1} "=";
printf{(i,j,k,l) in Eh: i = row and col >= j and col < l and xe1[i,j,k,l] = 0 and xe2[i,j,k,l] = 0} " "; printf{0..0: card({(i,j,k,l) in Eh: i = row and col >= j and col < l}) = 0} " "; } printf "\n"; for { col in cols } { printf{(i,j,k,l) in Ev: j = col and row >= i and row < k and xe1[i,j,k,l] = 1} "|"; printf{(i,j,k,l) in Ev: j = col and row >= i and row < k and xe2[i,j,k,l] = 1} '"'; printf{(i,j,k,l) in Ev: j = col and row >= i and row < k and xe1[i,j,k,l] = 0 and xe2[i,j,k,l] = 0} " "; /* No vertical edges: skip also a field */ printf{0..0: card({(i,j,k,l) in Ev: j = col and row >= i and row < k}) = 0} " "; printf " "; } printf "\n";}
data;
/* This is a difficult 25x25 Hashiwokakero. */param givens : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2425 := 1 2 . 2 . 2 . . 2 . 2 . . 2 . . . . 2 . 2 . 2 . 2 . 2 . 1 . . . . 2 . . . 4 . . 5 . 2 . . 1 . 2 . 2 . 1 3 2 . . 5 . 4 . . 3 . . . . . 1 . . 4 . 5 . 1 . 1 . 4 . . . . . . . . . . . 1 . 3 . . 1 . . . . . . . . 5 2 . . 6 . 6 . . 8 . 5 . 2 . . 3 . 5 . 7 . . 2 . . 6 . 1 . . . . . . . . . 1 . . 2 . . . . . 1 . . . 3 7 2 . . . . 5 . . 6 . 4 . . 2 . . . 2 . 5 . 4 . 2 . 8 . 2 . 2 . . . . . . . . . . . 3 . . 3 . . . 1 . 2 9 . . . . . . . . . . 4 . 2 . 2 . . 1 . . . 3 . 1 . 10 2 . 3 . . 6 . . 2 . . . . . . . . . . 3 . . . . . 11 . . . . 1 . . 2 . . 5 . . 1 . 4 . 3 . . . . 2 . 4 12 . . 2 . . 1 . . . . . . 5 . 4 . . . . 4 . 3 . . . 13 2 . . . 3 . 1 . . . . . . . . 3 . . 5 . 5 . . 2 . 14 . . . . . 2 . 5 . . 7 . 5 . 3 . 1 . . 1 . . 1 . 4 15 2 . 5 . 3 . . . . 1 . 2 . 1 . . . . 2 . 4 . . 2 . 16 . . . . . 1 . . . . . . . . . . 2 . . 2 . 1 . . 3 17 2 . 6 . 6 . . 2 . . 2 . 2 . 5 . . . . . 2 . . . . 18 . . . . . 1 . . . 3 . . . . . 1 . . 1 . . 4 . 3 . 19 . . 4 . 5 . . 2 . . . 2 . . 6 . 6 . . 3 . . . . 3 20 2 . . . . . . . . . 2 . . 1 . . . . . . 1 . . 1 . 21 . . 3 . . 3 . 5 . 5 . . 4 . 6 . 7 . . 4 . 6 . . 4 22 2 . . . 3 . 5 . 2 . 1 . . . . . . . . . . . . . . 23 . . . . . . . . . 1 . . . . . . 3 . 2 . . 5 . . 5 24 2 . 3 . 3 . 5 . 4 . 3 . 3 . 4 . . 2 . 2 . . . 1 . 25 . 1 . 2 . 2 . . . 2 . 2 . . . 2 . . . . 2 . 2 . 2 ;
end;
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