File size: 2,438 Bytes
1ca0407 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | """Edges vs triangles (Razborov) upper-bound curve (EinsteinArena edges-vs-triangles, MIT).
Each row is a probability vector over 20 parts (a complete 20-partite graphon); edge density
1 - sum w^2 ... (upstream uses S1^2 - S2 and S1^3 - 3 S1 S2 + 2 S3). Score = -(area under the slope-3
envelope + 10 * max gap) (maximize). Ported verbatim from the upstream verifier. Changes: at most
inst["max_rows"] rows (the description says m <= 500 but upstream did not enforce it); NaN/inf rejected."""
import numpy as np
from run import Invalid, float_matrix
def _densities(A):
S1 = A.sum(1)
S2 = (A ** 2).sum(1)
S3 = (A ** 3).sum(1)
return S1 ** 2 - S2, S1 ** 3 - 3 * S1 * S2 + 2 * S3
def _curve(ex, ty):
slope = 3.0
o = np.argsort(ex)
fx = np.concatenate(([0.0], ex[o], [1.0]))
fy = np.concatenate(([0.0], ty[o], [1.0]))
ux, inv = np.unique(fx, return_inverse=True)
my = np.full(len(ux), np.inf)
np.minimum.at(my, inv, fy)
fx, fy = ux, my
if len(fx) < 2:
return 5.0 / 6.0, 1.0
left = np.minimum.accumulate(fy - slope * fx)
right = np.minimum.accumulate(fy[::-1])[::-1]
area = 0.0
for i in range(len(fx) - 1):
x0, x1 = fx[i], fx[i + 1]
w = x1 - x0
if w <= 0:
continue
cap = min(float(right[i + 1]), 1.0)
yl, yr = slope * x0 + left[i], slope * x1 + left[i]
if yl >= cap:
area += cap * w
elif yr <= cap:
area += (yl + yr) * w / 2.0
else:
xh = (cap - left[i]) / slope
area += (yl + cap) * (xh - x0) / 2.0 + cap * (x1 - xh)
gaps = np.diff(fx)
idx = np.where((fx[:-1] >= 0.0) & (fx[:-1] < 1.0))[0]
return area, float(np.max(gaps[idx])) if idx.size else 0.0
def check(inst, ans):
w = ans.get("weights")
if not isinstance(w, list) or not 1 <= len(w) <= inst["max_rows"]:
raise Invalid(f"weights must be a list of 1..{inst['max_rows']} rows")
A = np.array(float_matrix(w, None, inst["parts"], "weights"))
if np.any(A < 0):
raise Invalid("weights must be non-negative")
s = A.sum(1)
if np.any(s < 1e-7):
raise Invalid("a row sums to (nearly) zero")
A = A / s[:, None]
e, t = _densities(A)
area, gap = _curve(e, t)
val = -(area + 10 * gap)
if not np.isfinite(val):
raise Invalid("non-finite score")
return float(val), {"area": float(area), "max_gap": gap}
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