Solve Hitori Puzzles

Haskell, 374 bytes

import Data.Array;import Data.List;r=range;p=partition
c(e,f)=p(\(b,p)->any(==1)[(b-d)^2+(p-q)^2|(d,q)<-e])f
n#g=[s|(o,(e:f))<-[p((==0).(g!))$indices g],
 null.fst$c(o,o),null.snd$until(null.fst)c([e],f),
 s<-case[((l,c),d)|((l,c),h)<-assocs g,h>0,
 d<-[filter((==h).(g!))$r((l,c+1),(l,n))++r((l+1,c),(n,c))],d/=[]]
 of[]->[g];((c,d):_)->n#(g//[(c,0)])++n#(g//[(c,0)|c<-d])]

Try it online!

Jelly, 62 bytes

Uses user202729's isConnected monadic link from another question.

---

FJṁa@µ«Ḋoµ€ZUµ4¡ÐLFQL<3
ḟ0ĠḊ€
¬T€œ&2\;Ç€FȦ
ZÇȯÇ_1Ŀ
2ḶṗLṗLa⁸ÇÞḢ

A full program printing a representation of a list of lists.
Works by brute force and is stupidly inefficient.

Try it online! - a 3 by 3, since it is too inefficient to run even a size 4 within the 60 second TIO limit!

How?

FJṁa@µ«Ḋoµ€ZUµ4¡ÐLFQL<3 - Link 1 isConnected? List of lists
...                     - 1 if connected 0 if not -- see linked answer in the header

ḟ0ĠḊ€ - Link 2, helperFor-AnyRepeatedValues: list
ḟ0    - filter out zeros
  Ġ   - group indices by value (i.e. [[indices of min],...,[indices of max]]
   Ḋ€ - dequeue €ach -- leaving a list of empty lists iff no repeated values
      -                 any remaining values are non-zero (1-based indexing in Jelly)

¬T€œ&2\;Ç€FȦ - Link 3, columnwiseAnyAdjacentZerosOrRowwiseAnyRepeatedValues: list of lists
¬            - logical not (convert all zeros to ones and all others to zeros)
 T€          - for €ach row get a list of truthy indexes (i.e. indexes of original zeros)
     2\      - pairwise reduction (i.e. for neighbouring rows) with:
   œ&        -   intersection (empty if no columnwise adjacent original zeros
             -                 any remaining values are non-zero due to 1-based indexing)
        Ç€   - call last link (1) as a monad for €ach row
       ;     - concatenate
          F  - flatten into a single list (empty iff no columnwise adjacent original zeros
             -                                   AND no rowwise repeated values)
           Ȧ - any and all (0 if empty [or contains any zero -- never] else 1)

ZÇȯÇ_1Ŀ - Link 4, validity check? list of lists
Z       - transpose
 Ç      - call last link (2) as a monad rowwiseAnyAdjacentZerosOrColumnwiseAnyRepeatedValues?
   Ç    - call last link (2) as a monad columnwiseAnyAdjacentZerosOrRowwiseAnyRepeatedValues?
  ȯ     - logical OR
     1Ŀ - call link 1 as a monad (isConnected?)
    _   - subtract
        - this yields -1 for valid, while it yields 0 or 1 if not.

2ḶṗLṗLa⁸ÇÞḢ - Main link: list of lists
2Ḷ          - lowered range of 2 -> [0,1]
   L        - length (number of rows in the input)
  ṗ         - Cartesian power (all lists of zeros and ones of length L)
     L      - length (number of rows in the input again)
    ṗ       - Cartesian power (all grids of zeros and ones of same shape as the input)
       ⁸    - the input
      a     - logical AND -- effectively uses each of the formed grids as a mask
         Þ  - sort by:
        Ç   -   last link (3) as a monad
          Ḣ - head
            - implicit print

APL (Dyalog Unicode), 133 bytes^SBCS

{q←{⊢/4 2⍴⍵}⌺3 3⋄g←⍵=⊂∪,⍵⋄⍵×~1⊃{((⌈/q b)⌈b<{2<≢∪0,,(⍵×⊢⌈⌈/∘q)⍣≡⍵×(⍴⍵)⍴1+⍳≢,⍵}¨~b∘⌈¨⊂⍤2∘.≡⍨⍳⍴b)(+/↑w<g×⌈.⌈⍨w×g)⌈w b←⍵}⍣≡×\(⌈/=∘⌽⍨q⍵)0}

Try it online!

My implementation of rule #4 (cells must form a single connected component) is rather wasteful, but still this passes all tests in about 10 seconds on TIO.

---

The overall algorithm: Store two boolean matrices b and w for cells that are certain to be black and white respectively. Initialise b as all-zero. Initialise w as 1 only for those cells that have opposite matching neighbours.

Repeat until b and w settle down:

• add to b cells that are on the same line (horizontal or vertical) and of the same value as a cell in w

• add to w the immediate neighbours of all cells in b

• add to w all cutpoints - cells whose removal would split the graph of non-black cells into multiple connected components

Finally, output not(b) multiplied by the original matrix.