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268 lines
9.2 KiB
268 lines
9.2 KiB
/* PBN, Paint-By-Numbers Puzzle */
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/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
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/* NOTE: See also the document "Solving Paint-By-Numbers Puzzles with
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GLPK", which is included in the GLPK distribution. */
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/* A paint-by-numbers puzzle consists of an m*n grid of pixels (the
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canvas) together with m+n cluster-size sequences, one for each row
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and column. The goal is to paint the canvas with a picture that
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satisfies the following constraints:
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1. Each pixel must be blank or white.
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2. If a row or column has cluster-size sequence s1, s2, ..., sk,
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then it must contain k clusters of black pixels - the first with
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s1 black pixels, the second with s2 black pixels, and so on.
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It should be noted that "first" means "leftmost" for rows and
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"topmost" for columns, and that rows and columns need not begin or
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end with black pixels.
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Example:
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1 1
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1 1
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2 1 1 1 1 1 2 3
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3 2 1 2 1 2 3 4 8 9
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3 6 # # # . # # # # # #
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1 4 # . . . . . # # # #
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1 1 3 . . # . # . . # # #
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2 . . . . . . . . # #
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3 3 . . # # # . . # # #
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1 4 # . . . . . # # # #
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2 5 # # . . . # # # # #
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2 5 # # . . . # # # # #
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1 1 . . . # . . . . . #
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3 . . # # # . . . . .
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(In Russia such puzzles are known as "Japanese crosswords".)
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References:
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Robert A. Bosch, "Painting by Numbers", 2000.
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<http://www.oberlin.edu/~math/faculty/bosch/pbn-page.html> */
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/*--------------------------------------------------------------------*/
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/* Main part based on the formulation proposed by Robert Bosch. */
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param m, integer, >= 1;
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/* the number of rows */
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param n, integer, >= 1;
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/* the number of columns */
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param row{i in 1..m, 1..n div 2}, integer, >= 0, default 0;
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/* the cluster-size sequence for row i (raw data) */
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param col{j in 1..n, 1..m div 2}, integer, >= 0, default 0;
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/* the cluster-size sequence for column j (raw data) */
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param kr{i in 1..m} := sum{t in 1..n div 2: row[i,t] > 0} 1;
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/* the number of clusters in row i */
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param kc{j in 1..n} := sum{t in 1..m div 2: col[j,t] > 0} 1;
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/* the number of clusters in column j */
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param sr{i in 1..m, t in 1..kr[i]} := row[i,t], integer, >= 1;
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/* the cluster-size sequence for row i */
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param sc{j in 1..n, t in 1..kc[j]} := col[j,t], integer, >= 1;
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/* the cluster-size sequence for column j */
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check{i in 1..m}: sum{t in 1..kr[i]} sr[i,t] <= n - (kr[i] - 1);
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/* check that the sum of the cluster sizes in each row is valid */
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check{j in 1..n}: sum{t in 1..kc[j]} sc[j,t] <= m - (kc[j] - 1);
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/* check that the sum of the cluster sizes in each column is valid */
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check: sum{i in 1..m, t in 1..kr[i]} sr[i,t] =
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sum{j in 1..n, t in 1..kc[j]} sc[j,t];
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/* check that the sum of the cluster sizes in all rows is equal to the
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sum of the cluster sizes in all columns */
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param er{i in 1..m, t in 1..kr[i]} :=
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if t = 1 then 1 else er[i,t-1] + sr[i,t-1] + 1;
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/* the smallest value of j such that row i's t-th cluster can be
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placed in row i with its leftmost pixel occupying pixel j */
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param lr{i in 1..m, t in 1..kr[i]} :=
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if t = kr[i] then n + 1 - sr[i,t] else lr[i,t+1] - sr[i,t] - 1;
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/* the largest value of j such that row i's t-th cluster can be
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placed in row i with its leftmost pixel occupying pixel j */
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param ec{j in 1..n, t in 1..kc[j]} :=
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if t = 1 then 1 else ec[j,t-1] + sc[j,t-1] + 1;
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/* the smallest value of i such that column j's t-th cluster can be
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placed in column j with its topmost pixel occupying pixel i */
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param lc{j in 1..n, t in 1..kc[j]} :=
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if t = kc[j] then m + 1 - sc[j,t] else lc[j,t+1] - sc[j,t] - 1;
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/* the largest value of i such that column j's t-th cluster can be
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placed in column j with its topmost pixel occupying pixel i */
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var z{i in 1..m, j in 1..n}, binary;
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/* z[i,j] = 1, if row i's j-th pixel is painted black
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z[i,j] = 0, if row i's j-th pixel is painted white */
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var y{i in 1..m, t in 1..kr[i], j in er[i,t]..lr[i,t]}, binary;
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/* y[i,t,j] = 1, if row i's t-th cluster is placed in row i with its
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leftmost pixel occupying pixel j
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y[i,t,j] = 0, if not */
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var x{j in 1..n, t in 1..kc[j], i in ec[j,t]..lc[j,t]}, binary;
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/* x[j,t,i] = 1, if column j's t-th cluster is placed in column j with
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its topmost pixel occupying pixel i
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x[j,t,i] = 0, if not */
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s.t. fa{i in 1..m, t in 1..kr[i]}:
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sum{j in er[i,t]..lr[i,t]} y[i,t,j] = 1;
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/* row i's t-th cluster must appear in row i exactly once */
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s.t. fb{i in 1..m, t in 1..kr[i]-1, j in er[i,t]..lr[i,t]}:
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y[i,t,j] <= sum{jp in j+sr[i,t]+1..lr[i,t+1]} y[i,t+1,jp];
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/* row i's (t+1)-th cluster must be placed to the right of its t-th
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cluster */
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s.t. fc{j in 1..n, t in 1..kc[j]}:
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sum{i in ec[j,t]..lc[j,t]} x[j,t,i] = 1;
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/* column j's t-th cluster must appear in column j exactly once */
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s.t. fd{j in 1..n, t in 1..kc[j]-1, i in ec[j,t]..lc[j,t]}:
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x[j,t,i] <= sum{ip in i+sc[j,t]+1..lc[j,t+1]} x[j,t+1,ip];
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/* column j's (t+1)-th cluster must be placed below its t-th cluster */
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s.t. fe{i in 1..m, j in 1..n}:
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z[i,j] <= sum{t in 1..kr[i], jp in er[i,t]..lr[i,t]:
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j-sr[i,t]+1 <= jp and jp <= j} y[i,t,jp];
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/* the double coverage constraint stating that if row i's j-th pixel
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is painted black, then at least one of row i's clusters must be
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placed in such a way that it covers row i's j-th pixel */
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s.t. ff{i in 1..m, j in 1..n}:
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z[i,j] <= sum{t in 1..kc[j], ip in ec[j,t]..lc[j,t]:
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i-sc[j,t]+1 <= ip and ip <= i} x[j,t,ip];
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/* the double coverage constraint making sure that if row i's j-th
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pixel is painted black, then at least one of column j's clusters
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covers it */
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s.t. fg{i in 1..m, j in 1..n, t in 1..kr[i], jp in er[i,t]..lr[i,t]:
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j-sr[i,t]+1 <= jp and jp <= j}: z[i,j] >= y[i,t,jp];
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/* the constraint to prevent white pixels from being covered by the
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row clusters */
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s.t. fh{i in 1..m, j in 1..n, t in 1..kc[j], ip in ec[j,t]..lc[j,t]:
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i-sc[j,t]+1 <= ip and ip <= i}: z[i,j] >= x[j,t,ip];
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/* the constraint to prevent white pixels from being covered by the
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column clusters */
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/* this is a feasibility problem, so no objective is needed */
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/*--------------------------------------------------------------------*/
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/* The following part is used only to check for multiple solutions. */
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param zz{i in 1..m, j in 1..n}, binary, default 0;
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/* zz[i,j] is z[i,j] for a previously found solution */
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s.t. fz{1..1 : sum{i in 1..m, j in 1..n} zz[i,j] > 0}:
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sum{i in 1..m, j in 1..n}
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(if zz[i,j] then (1 - z[i,j]) else z[i,j]) >= 1;
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/* the constraint to forbid finding a solution, which is identical to
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the previously found one; this constraint is included in the model
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only if the previously found solution specified by the parameter zz
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is provided in the data section */
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solve;
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/*--------------------------------------------------------------------*/
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/* Print solution to the standard output. */
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for {i in 1..m}
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{ printf{j in 1..n} " %s", if z[i,j] then "#" else ".";
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printf "\n";
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}
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/*--------------------------------------------------------------------*/
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/* Write solution to a text file in PostScript format. */
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param ps, symbolic, default "solution.ps";
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printf "%%!PS-Adobe-3.0\n" > ps;
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printf "%%%%Creator: GLPK (pbn.mod)\n" >> ps;
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printf "%%%%BoundingBox: 0 0 %d %d\n",
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6 * (n + 2), 6 * (m + 2) >> ps;
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printf "%%%%EndComments\n" >> ps;
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printf "<</PageSize [%d %d]>> setpagedevice\n",
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6 * (n + 2), 6 * (m + 2) >> ps;
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printf "0.1 setlinewidth\n" >> ps;
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printf "/A { 2 copy 2 copy 2 copy newpath moveto exch 6 add exch line" &
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"to\n" >> ps;
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printf "exch 6 add exch 6 add lineto 6 add lineto closepath } bind de" &
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"f\n" >> ps;
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printf "/W { A stroke } def\n" >> ps;
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printf "/B { A fill } def\n" >> ps;
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printf {i in 1..m, j in 1..n} "%d %d %s\n",
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(j - 1) * 6 + 6, (m - i) * 6 + 6,
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if z[i,j] then "B" else "W" >> ps;
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printf "%%%%EOF\n" >> ps;
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printf "Solution has been written to file %s\n", ps;
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/*--------------------------------------------------------------------*/
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/* Write solution to a text file in the form of MathProg data section,
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which can be used later to check for multiple solutions. */
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param dat, symbolic, default "solution.dat";
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printf "data;\n" > dat;
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printf "\n" >> dat;
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printf "param zz :" >> dat;
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printf {j in 1..n} " %d", j >> dat;
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printf " :=\n" >> dat;
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for {i in 1..m}
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{ printf " %2d", i >> dat;
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printf {j in 1..n} " %s", if z[i,j] then "1" else "." >> dat;
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printf "\n" >> dat;
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}
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printf ";\n" >> dat;
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printf "\n" >> dat;
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printf "end;\n" >> dat;
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printf "Solution has also been written to file %s\n", dat;
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/*--------------------------------------------------------------------*/
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/* The following data correspond to the example above. */
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data;
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param m := 10;
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param n := 10;
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param row : 1 2 3 :=
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1 3 6 .
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2 1 4 .
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3 1 1 3
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4 2 . .
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5 3 3 .
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6 1 4 .
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7 2 5 .
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8 2 5 .
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9 1 1 .
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10 3 . .
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;
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param col : 1 2 3 4 :=
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1 2 3 . .
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2 1 2 . .
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3 1 1 1 1
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4 1 2 . .
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5 1 1 1 1
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6 1 2 . .
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7 2 3 . .
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8 3 4 . .
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9 8 . . .
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10 9 . . .
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;
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end;
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