Can he give the policeman money so that he will not fine him for breaking traffic rules?

Stopping at the traffic lights is obligatory, because they were set up in the public interest, to regulate the flow of traffic and protect lives and property. If people drove on the streets ignoring these signals, that would result in a great deal of trouble for them, as is well known. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) issued a fatwa stating that it is not permissible to run a red light. He regarded that as coming under the heading of obeying the authorities, which is obligatory, because Allaah says (interpretation of the meaning):  

“O you who believe! Obey Allaah and obey the Messenger (Muhammad صلى الله عليه وسلم), and those of you (Muslims) who are in authority”

[al-Nisa’ 4:59]. 

Liqaa’aat al-Baab il-Maftooh (3/178), question no. 1265 

As for making the one who breaks these laws pay a certain fine, this comes under the heading of financial disciplinary punishments, which is permissible according to many scholars, such as Abu Yoosuf, the companion of Abu Haneefah; it is also the earlier view of al-Shaafa’i and was the view of some of the Maalikis; it was also the view favoured by Ibn Taymiyah and Ibn al-Qayyim (may Allaah have mercy on them all). 

Ibn al-Qayyim said: With regard to ta’zeer (disciplinary punishments) in the form of financial penalties, this is also prescribed in certain cases according to the view of Maalik and Ahmad, and one of the two views of al-Shaafa’i. The Sunnah of the Messenger of Allaah (peace and blessings of Allaah be upon him) and his companions also mentions that in some cases, such as when the Prophet (peace and blessings of Allaah be upon him) ordered that the vessels and containers of wine be broken; when he ordered ‘Abd-Allaah ibn ‘Umar to burn the two garments that were dyed with safflower; and when on the day of Khaybar he ordered that the pots in which the flesh of domesticated donkeys had been cooked be broken, then they asked him for permission to wash them and he gave them permission. This indicates that both things are permissible, because the penalty is not necessarily by breaking. Another example is when he demolished the mosque of the hypocrites; when he burned the belongings of the one who had stolen from the war booty before it had been shared out; when he took half of the wealth of the one who had withheld zakaah, on the orders of Allaah; when he ordered the one who was wearing a ring of gold to throw it away, and he did so, and no one touched it; when he cut down the palm trees of the Jews to annoy them; when ‘Umar and ‘Ali burned down places in which alcohol was sold; when ‘Umar burned the castle of Sa’d ibn Abi Waqqaas when he hid from the people he had been put in charge of.  These cases are all saheeh and well known, and it cannot be claimed that they have been abrogated. The one who says that financial penalties have been abrogated and that this is general in application has gone beyond the views of the imams in quoting and reaching conclusions. Most of these issues are acceptable according to the madhhab of Ahmad and others, and many of them are acceptable according to Maalik. The fact that the Rightly-Guided Caliphs and the senior Sahaabah did these things after the death of the Prophet (peace and blessings of Allaah be upon him) also shows that the claim that the ruling has been abrogated is false. Those who say that it has been abrogated have no evidence from Qur’aan or Sunnah or scholarly consensus to support their claims. End quote from al-Turuq al-Hukamiyyah (p. 224). 

According to the view that it is permissible to impose financial penalties, it is not permissible to try to get out of paying this fine, and the meal that was offered to the policeman is a haraam bribe, because it was given to the policeman so that he would waive something that he was obliged to pay. 

As for the permissible bribe, that is the one that a person is forced to give in order to ward off injustice from himself when he is being wronged, and it can only be given in such cases. 

See also the answer to question no. 25758. 

And Allaah knows best.