unit cost gap is not better

main.tex

@@ -343,7 +343,7 @@ Can we show that the gap is 0 or much smaller than 2?
 \begin{enumerate}
 \item One cannot do better than $b\lambda^*$ for general costs.
 There are examples (a 4-vertex path with parallel edges) where the gap is almost $b\lambda^*$.\footnote{see \url{https://gitea.talldoor.uk/sxlxc/edge_conn_interdiction/src/branch/master/gap.py}}
-\item Unit cost. We can assume WLOG that $|C^*|>b$ and that $F^*$ is the set of $b$ edges in $C^*$ with largest weights. By the complementary slackness condition, $(C^{LD},F^{LD})$ is optimal for connectivity interdiction IP. Thus we can see the gap is $1$.
+\item Unit cost. There is still a gap between $L(\lambda^*)$ and $w(C^*\setminus F^*)$.\footnote{see \url{https://gitea.talldoor.uk/sxlxc/edge_conn_interdiction/src/branch/master/plot.py}}
 \end{enumerate}
 
 \end{remark}

plot.py

@@ -0,0 +1,62 @@
+import matplotlib.pyplot as plt
+import numpy as np
+
+
+def plot_linear_functions(list1, list2, list3, b, x_range=(0, 10)):
+    """
+    Plots linear functions y = Cx + D where:
+    C = b - k (k elements removed)
+    D = sum of remaining elements
+    k ranges from 0 to b.
+    """
+    datasets = [list1, list2, list3]
+    # Distinct colors for the 3 sets
+    colors = ['#E63946', '#457B9D', '#1D3557']
+    labels = ['List 1', 'List 2', 'List 3']
+
+    plt.figure(figsize=(12, 8))
+    x = np.linspace(x_range[0], x_range[1], 100)
+
+    for i, current_list in enumerate(datasets):
+        # Sort descending so the first k elements are the 'top k'
+        sorted_list = sorted(current_list, reverse=True)
+
+        for k in range(len(sorted_list) + 1):
+            # C is the fixed number b minus the number of elements removed
+            C = k-b
+
+            # Remove top k elements (if k > list length, remaining is empty)
+            remaining = sorted_list[k:] if k < len(sorted_list) else []
+
+            # D is the sum of the remaining elements
+            D = sum(remaining)
+
+            y = C * x + D
+
+            # Plot the line; label only once per list for a clean legend
+            line_label = labels[i] if k == 0 else ""
+            if C == 0:
+                plt.plot(x, y, color=colors[i], alpha=1,
+                         label=line_label, linestyle='-')
+            else:
+                plt.plot(x, y, color=colors[i], alpha=0.6,
+                         label=line_label, linestyle='--')
+
+    plt.title(
+        f"2D Linear Functions: $y = (b-k)x + sum(rem)$ for $0 ≤ k ≤ b$ ($b={b}$)")
+    plt.xlabel("x")
+    plt.ylabel("y")
+    plt.grid(True, linestyle='--', alpha=0.7)
+    plt.legend()
+    plt.savefig('linear_plot.png')
+    plt.show()
+
+
+# --- Example Usage ---
+# Replace these with your actual lists and b value
+L1 = [45, 14, 7, 7, 4, 2]
+L2 = [20, 15, 10, 5, 4, 2]
+L3 = [5, 5, 5, 5, 5, 5]
+fixed_b = 2
+
+plot_linear_functions(L1, L2, L3, fixed_b)