Is the pizza fair?

Snails, 129

Prints 1 for a fair pizza and 0 for an unfair pizza.

&
={(t\Dt\Et\Ct\M),!(t.}{(o\D)+l^D,=~u{^D=(r^D,~},~|o\E`+l^E,=~u{^E=(r^E,~},~|o\C`+l^C,=~u{^C=(r^C,~},~|o\M`+l^M,=~u{^M=(r^M,~},~

Expanded version:

&
={ (t\Dt\Et\Ct\M), !(t.)}   {
(o\D)+ l^D,=~ u{^D=(r^D,~)}, ~ |
(o\E)+ l^E,=~ u{^E=(r^E,~)}, ~ |
(o\C)+ l^C,=~ u{^C=(r^C,~)}, ~ |
(o\M)+ l^M,=~ u{^M=(r^M,~)}, ~

& means that the pattern must match at all locations on the grid. The first line checks for an equal number of each of E, D, M, C. it uses the teleport instruction t, which is a great way to make programs with factorial complexity. If an input has unequally sized slices with several units for each of the 4 mods, the program will more or less hang forever. After that, there is a check for a contiguous path to the top-left instance of whichever letter the pattern started on.

CJam, 93

qN/_z,:W;s:A,,:B_{{{_B=_@-}g}%$}:F;{a+_Af=)#{F~B\@t:B}|;}:U;W>{_W-U}/{W%},{_(U}/BFe`0f=_1<4*=

Try it online

This is ridiculously long because CJam doesn't (yet) have built-in flood-fill or union-find. I implemented union-find in the program.

Explanation:

qN/_         read input, split into lines and duplicate
z,:W;        transpose, get length (original width) and store in W
s:A          convert to string (without newlines) and store in A
,,           make an array [0..n-1] (n = pizza size)
:B_          store in B (initial structure) and duplicate (will be used in 2 loops)
{…}:F;       define function F ("Find" for multiple indices and sort)
  {…}%       for each value (x)
    {…}g     do…while
      _B=    duplicate x and get B[x]
      _@-    leave a B[x] on the stack and calculate B[x] - x
              if non-zero, repeat the loop with B[x]
  $          sort the results
{…}:U;       define function U ("Union" for 2 indices)
  a+         make an array of the 2 indices
  _Af=       get the corresponding letters from A
  )#         check if the letters are different
  {…}|       if not, execute…
    F~       call F on the array and dump the 2 results on the stack
    B\@t     join the sets - B[bigger index] = smaller index
    :B       store back in B
  ;          pop the last value (either array if indices or B)
W>           remove the first row of indices
{…}/         for each index
  _W-        duplicate and subtract W ("go up")
  U          call U to join sets if they match
{W%},        remove the first column of indices
{…}/         for each index
  _(         duplicate and decrement ("go left")
  U          call U to join sets if they match
BF           call F on B, to get the final sets and sort
e`           RLE encoding
0f=          keep only the repetition counts
_1<4*=       check if it's the first value (if any) repeated 4 times

Pyth, 53 bytes

!f-lJs.z*4lu&G{smfqT@JY@UJ+Ld[Z1_1Klh.z_K)G]xJT)"CDEM

Demonstration

This is essentially a flood fill for each letter, followed by a check that all of the resulting sets are of the appropriate size.

To flood-fill, it starts with the upper-left-most occurence of each letter, then generates all neighbors of locations found so far, filters for locations with the right letter, and repeats until the set stops changing.