Here is a Python function that takes the inputs you described and returns the minimum cost or $-1$ if it's not possible:$-$

def minimize$\_$cost$\_$to$\_$generate$\_$string$($S$,$ words, K$,$ costs$)$:

# Dictionary to store the minimum cost for each substring

dp $=\{\}$

def min$\_$cost$\_$to$\_$generate$($substring$)$:

# Base case: an empty substring can be generated with $0$ cost

if not substring:

return $0$

# Checking if the cost for the current substring is already calculated

if substring in dp:

return dp$[$substring$]$

# Initializing the cost to $-1($indicating it's not possible to generate the substring$)$

min$\_$cost $=-1$

# Trying to generate the substring by concatenating words

for i in range$($K$)$:

word $=$ words$[$i$]$

cost $=$ costs$[$i$]$

if substring.startswith$($word$)$:

# Recursively calculating the cost for the remaining substring

remaining$\_$cost $=$ min$\_$cost$\_$to$\_$generate$($substring$[$len$($word$)$:$])$

# If it's possible to generate the remaining substring, update the minimum cost

if remaining$\_$cost $!=-1$:

current$\_$cost $=$ cost $+$ remaining$\_$cost

if min$\_$cost $==-1$ or current$\_$cost $<$ min$\_$cost:

min$\_$cost $=$ current$\_$cost

# Saving the calculated cost in the dictionary

dp$[$substring$]=$ min$\_$cost

return min$\_$cost

# Calling the recursive function to find the minimum cost for the entire string S

result $=$ min$\_$cost$\_$to$\_$generate$($S$)$

return result

# Example usage:

S $=$ "abcab"

words $=["$ab$","$c$",$ "abc"$]$

K $=3$

costs $=[1,2,3]$

result $=$ minimize$\_$cost$\_$to$\_$generate$\_$string$($S$,$ words, K$,$ costs$)$

print$($result$)$

Explanation

Here the dynamic programming is used to find the minimum cost to generate the given string