as2: task 1 a and b

assignment-2/main.typ

@@ -1,5 +1,6 @@
 #import "@preview/simple-ntnu-report:0.1.2": ntnu-report, un
 #import "@preview/zero:0.5.0": num
+#import "@preview/physica:0.9.8": *
 
 #let text-size = 12pt
 #let white = rgb("#F8F8F2")
@@ -41,17 +42,63 @@
 )
 
 #set text(size: text-size, fill: white)
+#set math.equation(numbering: none)
 
 #show heading: set text(style: "italic")
-#show heading: it => {box(
-    height: 14pt,
-    width: 90%,
-    outset: 5%,
+#show heading.where(level: 2): it => {box(
+    height: text-size * 1.3,
+    width: 100%,
+    outset: 2.5%,
     radius: 25%,
     fill: green,
     align(horizon, text(fill: black, it.body))
 )}
+#show heading.where(level: 3): it => {
+  box(strong(it.body) + h(0.5em))
+}
+
+#let ll(l, t) = text(cyan)[#link(l)[#t]]
 
 == task 1
+this sounds like the #ll("https://en.wikipedia.org/wiki/Vertex_cover")[vertex cover problem], but with additional vertex weights to minimize.
 
-hello world
+=== a)
+recall the general key components of a linear program
+- problem input vector $x$
+- coefficient vector $c$
+- constraint matrix $A$
+- target vector $b$
+such that we minimize $c^TT x$ with respect to $A x >= b$.
+
+here we can interpret the input vector $x$ as a binary vector where the $i$-th bit encodes the inclusion of a given node $v_i in V$ in the subset $C$. $x$ has as many bits as there are vertices.
+
+furthermore, the vector $c$ represents the weights of the vertices such that $
+c^TT x = sum_(v in V) c_v x_v = sum_(v in C) c_v
+$ becomes the minimization goal.
+
+we can express the binary string $x$ through constraints
+- $x_u + x_v >= 1 quad forall (u, v) in E$
+- $x_v in {0, 1} quad forall v in V$
+
+the former constraint ensures that every edge is covered by at least one endpoint.
+
+bringing it all together
+$
+    min sum_(v in V) c_v x_v \
+    "s.t." quad x_u + x_v >= 1 quad forall (u, v) in E \
+    x_v in {0, 1} quad forall v in V
+$
+is our complete linear integer program.
+
+we have implicitly defined $A$ to be the $|E| times |V|$-matrix that has $1$-entries for vertices $v$ where $e = (u, v)$. it is zero otherwise. 
+
+=== b)
+by relaxing the linear integer program we let $0 <= x_v <= 1$. this can be shortened to saying $x_v >= 0$ since the first constraint forces $x_v <= 1$. requiring a positive value is a common restriction on linear programs and allows us to take the dual. 
+
+we then obtain the dual program
+$
+    max bold(1)^TT y = sum_(e in E) y_e \
+    "s.t." quad A^TT y <= c \
+    <=> sum_(v in e) y_e <= c_v \
+    quad forall v in V, quad y >= 0
+$