as2: finish task 1

assignment-2/main.typ

@@ -1,6 +1,8 @@
 #import "@preview/simple-ntnu-report:0.1.2": ntnu-report, un
 #import "@preview/zero:0.5.0": num
 #import "@preview/physica:0.9.8": *
+#import "@preview/algorithmic:1.0.7"
+#import algorithmic: style-algorithm, algorithm-figure
 
 #let text-size = 12pt
 #let white = rgb("#F8F8F2")
@@ -13,6 +15,14 @@
 #let purple = rgb("#BD93F9")
 #let pink = rgb("#FF79C6")
 
+#show: style-algorithm.with(
+  hlines: (
+    grid.hline(stroke: 0.5pt + white),
+    grid.hline(stroke: 0.5pt + white),
+    grid.hline(stroke: 0.5pt + white)
+  )
+)
+
 #set page(fill: black)
 
 #set text(size: text-size, fill: white)
@@ -102,3 +112,48 @@ $
     <=> sum_(v in e) y_e <= c_v \
     quad forall v in V, quad y >= 0
 $
+
+=== c)
+to construct a primal-dual algorithm to approximate this problem, we use the formulations given previously.
+
+
+#algorithm-figure(
+    "Primal-Dual Weighted Vertex Cover Approximation",
+    vstroke: .5pt + white, {
+        import algorithmic: *
+        Procedure(
+            "Vertex-Cover",
+            ("G", "c"), {
+                Comment[initialize primal ($x$) and dual ($y$) variables]
+                For($v in V, e in E$,
+                    Assign[$y_e$][$0$],
+                    Assign[$x_v$][$0$]
+                )
+                LineBreak
+                While([there exists an uncovered edge $e = (u, v)$], {
+                    Comment[calculate minimum slack]
+                    Assign[$"slack"_u$][$c_u - sum_(e' in delta(u)) y_(e')$]
+                    Assign[$"slack"_v$][$c_v - sum_(e' in delta(v)) y_(e')$]
+                    Assign[$alpha$][$min("slack"_u, "slack"_v)$]
+                    LineBreak
+                    Comment[increase dual variable for edge $e$]
+                    Assign[$y_e$][$y_e + alpha$]
+                    LineBreak
+                    Comment[Update primal if constraint becomes tight]
+                    If([$sum_(e' in delta(u)) y_(e') = c_u$],
+                        Assign[$x_u$][$1$])
+                    If([$sum_(e' in delta(v)) y_(e') = c_v$],
+                        Assign[$x_v$][$1$])
+                })
+                Return[$C = {v in V | x_v = 1}$]
+            },
+        )
+    }
+)
+
+where $delta(u)$ means the set of all edges going out of vertex $u$.
+
+== other
+
+the rest of the tasks are attached or can be found at
+https://git.pvv.ntnu.no/frero-uni/TDT4125/src/branch/main/assignment-2