Add solution to 2025-12-10

Author: fuglede

2025/day10/solutions.py

@@ -0,0 +1,59 @@
+"""
+The solution below is really one in which the solution for part 2 has been
+retroactively applied to part 1 as well, simply because it was a fun thing to do.
+
+Really, part 1 is likely always best solved by a combinatorial/path based solution like
+(bidirectional) BFS or an SAT/SMT solver like Z3, whereas part 2 is very naturally
+expressed as an integer linear program.
+
+The solution below takes that part 2 solution, and extends it by also formulating
+part 1 as an integer linear program. Here's how that works: In both parts, we want to
+solve a linear equation 𝐴𝑥 = 𝑏 such that 𝑥 is minimal in some sense. In part 1, we
+are solving over 𝔽₂ = {0, 1} and require that 𝑥 has minimal Hamming weight, and in
+part 2, we require that 𝑥 consists of positive integers and has minimal sum. Integer
+programming solvers like the HiGHS-based solver in SciPy that we use below like
+integers better but can be forced to work over the binaries by adding additional
+variables. Concretely, for part 1, instead of solving 𝐴𝑥 = 𝑏 over 𝔽₂, we instead
+allow 𝑥 to be an integer, and introduce new integer variables 𝑡 and solve
+𝐴𝑥 − 2𝑡 = 𝑏, still just minimizing the sum of 𝑥. This works because 2𝑡 vanishes over
+𝔽₂, and because any optimal integer solution will values of 𝑥 in {0, 1}. In other
+words, to solve part 1, we take our part 2 solution and simply add a handful of useful
+auxiliary variables to encode the fact that the matrix equation is now modulo 2.
+
+While we're at it, note that part 1 is actually the well-known “syndrome decoding”
+problem, also known as “maximum-likelihood decoding”.
+"""
+
+import numpy as np
+from scipy.optimize import linprog
+
+
+with open("input") as f:
+    ls = f.read().strip().split("\n")
+
+tasks = []
+for l in ls:
+    toggles, *buttons, counters = l.split()
+    toggles = [x == "#" for x in toggles[1:-1]]
+    moves = [set(map(int, b[1:-1].split(","))) for b in buttons]
+    counters = list(map(int, counters[1:-1].split(",")))
+    tasks.append((toggles, moves, counters))
+
+
+def solve(goal, moves, part1):
+    n, m = len(moves), len(goal)
+    c = [1] * n
+    A_eq = [[i in move for move in moves] for i in range(m)]
+    bounds = [(0, None)] * n
+    if part1:
+        c += [0] * m
+        A_eq = np.hstack([A_eq, -2 * np.eye(m)])
+        bounds += [(None, None)] * m
+    return linprog(c, A_eq=A_eq, b_eq=goal, bounds=bounds, integrality=True).fun
+
+
+# Part 1
+print(sum(solve(goal, moves, True) for goal, moves, _ in tasks))
+
+# Part 2
+print(sum(solve(goal, moves, False) for _, moves, goal in tasks))