Although favored by his patron goddess Athena, Odysseus must leave his home to fight in the Trojan War. He is condemned to further years of wandering because he blinds Poseidon's son the Cyclops. As a wanderer, he is forever a strange man in a strange place, always looking for home. The Greeks greatly valued home as a physical place and home as an existential concept -- the Greeks were known for calling all non-Greeks barbarians, by virtue of not being Greeks, and within Greece there were frequent civil wars between different city-states because of local pride. However, hospitality towards strangers was demanded by the Greek sense of fear of being placeless. "Xenia is the Greek relationship between two people from different regions. This [value] allowed for the members of the relationship to safely travel into the other member's territory and receive a place to stay and something to eat" (Biggs et al. 2009). This is one reason why Penelope's suitors in the "Odyssey" are seen as especially brutal, because they violate the principles of being a good guest to a hospitable host.
The Indian hero Rama is almost always shown with a bow over his shoulder, and Odysseus is famed for his skill as an archer (Murthy 2004). However, unlike Odysseus, Rama only attacks when provoked and only engages in aggressive behavior when others are under threat, such as his beloved wife Sita by the evil monarch Ravana. Although the suitor's eating his food and wooing his wife could be considered provocation, Odysseus is overall a more aggressive figure than the Hindu prince deity. Also, Rama's willing submission to the will of others stands in profound contrast to Odysseus' proud individualism and command of his men and troops. While Odysseus alone survives his wanderings, largely as a result of his own cleverness and the help of Athena, Rama is far more dependant upon others for assistance, demonstrating the value of interconnectedness in Indian society.
Gilgamesh, in contrast to both good rulers Odysseus and Rama is actually oppressive towards his people at the beginning of the tale. More so than the other heroes, he has much to learn about the nature of human compassion and existence. While it is true that like Gilgamesh, Odysseus often falls afoul of the gods, this is more due to his cleverness than his brute strength.