well.. giving my own opinion... i m not sure there is a study that support such thing and its impact on the society as a whole and to children and how much it would actually benefits vs harm relationships in different areas of a relationship structure.
"There are many rapes, murders, crimes due to sex and unbalance laws in India."
Even tho what u suggested may help temporary, but it is not a permanent thing.. it is still something that deals only with the results (rapes murders) and not actually dealing with the cause of such results. in other words, It will not fix the unbalanced laws in india and will not help in fixing the deep motives behind doing actions such as rapes and murders...
Also, I m not sure the trust can be built through sex. it can be improved but if there was no trust, i m not sure how far it can be created through sex. you can see a lot of trust issue among couples who have sex everyday or even in an open relationship.
In my opinion, trust is an emotional state and it is built over a period of time, not necessary long or short.. but it is not built with just merely having sex in bed.. and since u are speaking of big number of partners, it will require a lot of time.
Generally speaking, maintaining a relationship made of two isn't that easy, due to different views, opinions, personality characteristics...etc. what about having 3 or even more in a relationship... each one imposing his views, opinion, lies, misunderstanding and groups start to polarize...etc.
Especially when having kids... each one will try to impose his views on how it should be raised and give different advice due to different experiences in life, what they think they are wrong, what is right...etc.
Polygamy has its benefits in some areas but i m not sure it is a solution for murders, rapes and trust...