Jinbei Tsukishima is a warrior who aims to be best, under the belief that a weak warrior isn't a warrior at all. He was scouted out by a representative to join the Mushibugyo and told to go to Edo with the instructions that he'd knew what to do once he got there. 

The action story is set in Edo-era Tokyo in the early 1700s. The eighth shogun Tokugawa Yoshimune ordered boxes to be installed so commoners can submit their thoughts. After hearing their voices, the shogun creates a new magistrate unit—Mushibugyou—to protect the people in the heart of the city. Each member is a specialist with unique fighting skills. 
(Source: Anime News Network)

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In Feudal Japan, the people of Edo are under siege by giant insects that ravage the land. The people desperately beg the government to do something about it. Thus the Insect Magistrate Office is established, gathering strong warriors to defend against the onset of pests. 
Jinbee Tsukishima is a young man following the belief that a weak warrior isn't a warrior at all and strives to be a master swordsman just like his father. To atone for a horrific incident that occurred at his fault, Jinbee seeks to take his father's place as a member of the Insect Magistrate Office. On his journey there, he meets the lovely Haru, a young woman who helps manage her family's tea house, and is forced to put his sword to good use in saving her from a grisly fate. This act of bravery earns him a spot in the Insect Magistrate. Will this rookie exterminator be able to rid the land of the horde of insects swarming in?

OVAs bundled with various volumes of the manga.

The people have demanded change; Criterion boxes were created for the purpose of the people to leave their complaints in it. The result of this were two changes: the first being the creation of "Koishigawa Youjoujo" for the people, the second being an inner-city magistrate office, known as the Mushibugyou. 
Tsukishima Jinbee is a warrior who aims to be best, under the belief that a weak warrior isn't a warrior at all. He was scouted out by a representative to join the Mushibugyou and told to go to Edo with the instructions that he'd know what to do once he got there. Shortly after arriving at Edo, he hears a scream and goes to check it out... to find a giant spider-like creature devouring some citizens of Edo. Being a warrior, there's no way he could stand this...
 
Note: Chapter count includes 1 extra.