Growing Trees
Description
You are given an undirected connected simple graph with $n$ nodes and $m$ edges, where edge $i$ connects node $u_i$ and $v_i$, with two positive parameters $a_i$ and $b_i$ attached to it. Additionally, you are also given an integer $k$.
A non-negative array $x$ with size $m$ is called a $k$-spanning-tree generator if it satisfies the following:
- Consider the undirected multigraph with $n$ nodes where edge $i$ is cloned $x_i$ times (i.e. there are $x_i$ edges connecting $u_i$ and $v_i$). It is possible to partition the edges of this graph into $k$ spanning trees, where each edge belongs to exactly one spanning tree$^\dagger$.
The cost of such array $x$ is defined as $\sum_{i = 1}^m a_i x_i^2 + b_i x_i$. Find the minimum cost of a $k$-spanning-tree generator.
$^\dagger$ A spanning tree of a (multi)graph is a subset of the graph's edges that form a tree connecting all vertices of the graph.
Each test contains multiple test cases. The first line contains an integer $t$ ($1 \le t \le 500$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains three integers $n$, $m$, and $k$ ($2 \le n \le 50, n - 1 \le m \le \min(50, \frac{n(n - 1)}{2}), 1 \le k \le 10^7$) — the number of nodes in the graph, the number of edges in the graph, and the parameter for the $k$-spanning-tree generator.
Each of the next $m$ lines of each test case contains four integers $u_i$, $v_i$, $a_i$, and $b_i$ ($1 \le u_i, v_i \le n, u_i \neq v_i, 1 \le a_i, b_i \le 1000$) — the endpoints of the edge $i$ and its two parameters. It is guaranteed that the graph is simple and connected.
It is guaranteed that the sum of $n^2$ and the sum of $m^2$ over all test cases does not exceed $2500$.
For each test case, output a single integer: the minimum cost of a $k$-spanning-tree generator.
Input
Each test contains multiple test cases. The first line contains an integer $t$ ($1 \le t \le 500$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains three integers $n$, $m$, and $k$ ($2 \le n \le 50, n - 1 \le m \le \min(50, \frac{n(n - 1)}{2}), 1 \le k \le 10^7$) — the number of nodes in the graph, the number of edges in the graph, and the parameter for the $k$-spanning-tree generator.
Each of the next $m$ lines of each test case contains four integers $u_i$, $v_i$, $a_i$, and $b_i$ ($1 \le u_i, v_i \le n, u_i \neq v_i, 1 \le a_i, b_i \le 1000$) — the endpoints of the edge $i$ and its two parameters. It is guaranteed that the graph is simple and connected.
It is guaranteed that the sum of $n^2$ and the sum of $m^2$ over all test cases does not exceed $2500$.
Output
For each test case, output a single integer: the minimum cost of a $k$-spanning-tree generator.
4
5 5 1
4 3 5 5
2 1 5 7
2 4 6 2
5 3 3 5
2 5 2 9
5 5 3
4 3 5 5
2 1 5 7
2 4 6 2
5 3 3 5
2 5 2 9
2 1 10000000
1 2 1000 1000
10 15 10
7 1 7 6
5 8 6 6
4 8 2 2
4 3 10 9
10 8 3 4
4 6 6 1
5 4 1 3
9 3 4 3
8 3 9 9
7 5 10 3
2 1 3 4
6 1 6 4
2 5 7 3
10 7 2 1
8 2 6 8
38
191
100000010000000000
2722
Note
In the first test case, a valid $1$-spanning-tree generator is $x = [1, 1, 1, 1, 0]$, as indicated by the following figure. The cost of this generator is $(1^2 \cdot 5 + 1 \cdot 5) + (1^2 \cdot 5 + 1 \cdot 7) + (1^2 \cdot 6 + 1 \cdot 2) + (1^2 \cdot 3 + 1 \cdot 5) + (0^2 \cdot 4 + 0 \cdot 9) = 38$. It can be proven that no other generator has a lower cost.
The $1$-spanning-tree partition of $x = [1, 1, 1, 1, 0]$ In the second test case, a valid $3$-spanning-tree generator is $x = [2, 3, 2, 2, 3]$, as indicated by the following figure. The cost of this generator is $(2^2 \cdot 5 + 2 \cdot 5) + (3^2 \cdot 5 + 3 \cdot 7) + (2^2 \cdot 6 + 2 \cdot 2) + (2^2 \cdot 3 + 2 \cdot 5) + (3^2 \cdot 4 + 3 \cdot 9) = 191$. It can be proven that no other generator has a lower cost.
The $3$-spanning-tree partition of $x = [2, 3, 2, 2, 3]$