Boolean Game BOOLGAME Solution

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Boolean Game BOOLGAME April Long Challenge 2021

Consider N binary variables x1,x2,…,xN. For each valid i, the i-th of these variables can be xi=0 or xi=1; therefore, there are 2N possible assignments of values to the variables. For each valid i, setting xi=1 gives you score gi.

In addition, there are M special intervals (numbered 1 through M). For each valid i, the i-th interval is [ui,vi] and if xui=xui+1=…=xvi=1, then your score increases by di.

Note that both gi and di can be negative (setting more variables to 1 can decrease your score), and your score can also be negative. Formally, the score of an assignment of values to the binary variables is
∑i=1Ngi⋅xi+∑i=1Mdi⋅∏j=uivixj.
Find the K highest scores among all assignments on these variables. Formally, if we computed the scores of all 2N assignments and sorted them in non-increasing order, you should find the first K of these values.

Input
The first line of the input contains a single integer T denoting the number of test cases. The description of T test cases follows.
The first line of each test case contains three space-separated integers N, M and K.
The second line contains N space-separated integers g1,g2,…,gN.
M lines follow. For each valid i, the i-th of these lines contains three space-separated integers ui, vi and di.
Output
For each test case, print a single line containing K space-separated integers ― the K highest scores of assignments on the binary variables, in decreasing order.

Constraints
1≤T≤10
1≤N≤60
1≤M≤min(N⋅(N−1)/2,1,000)
1≤K≤min(2N,200)
|gi|≤109 for each valid i
1≤ui<vi≤N for each valid i
for each valid i and j (i≠j), ui≠uj or vi≠vj
|di|≤109 for each valid i
the sum of N over all test cases does not exceed 60
Subtasks
Subtask #1 (5 points): the sum of N over all test cases does not exceed 18
Subtask #2 (20 points): K=1
Subtask #3 (75 points): original constraints

Example Input
1
4 2 3
-4 -2 5 2
1 3 0
1 4 -3
Example Output
7 5 5
Explanation
Example case 1: The best assignment is x=(0,0,1,1), with score 7. The second and third best assignments are (0,0,1,0) and (0,1,1,1), each with score 5.

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