That map is skewed because it references pricing for a 2-bedroom apartment and uses "market rent" which is almost always higher than the actual amount one pays (it's how apartments trick you into thinking you're getting a deal).
Here's a nice complex near my office:
Dana Point Apartments in Dallas, TX | Apartments.com
If I'm single with part-time income from two jobs totaling 40 hours a week, I have no need for anything larger than the 1-bedroom unit.
So here's the total:
$468/month for rent
$45/month for a no-contract cell phone
$40/month for internet (no cable..but why? I can watch TV online for free!)
$90/month for electricity
$150/month for food and toiletries
Total: $793
Now...I could walk to work from that apartment if I were working at many of the local fast food places (about the only places that pay minimum wage), but let's assume I have to take public transit to work. The monthly pass is $80 and gives me access to both the bus and the rail system that covers most of the metroplex.
So that takes me to $873 a month. We'll add another $20 for water/trash and we're at $893.
So minimum wage is $7.25 in Texas. At 40 hours a week, that would come to $290.00/week, or $1256.67 a month [($290 *52)/12]. Or about $1,005.34 a month after taxes.
I'd be left with $112.34 a month to spend on other needs.
Starbucks (about 3 blocks from the apartment) starts you off at around $8.59. If I do that 40 hours a week, I'm at $343.60/week, or $1,488.93 a month ($1,191.15 after taxes). That leaves me with $298.15 a month for other needs.
Wal-Mart starts you off at around $8.50.
Target starts overnight stockers at around $9.50.
The average Olive Garden waitstaff makes between $18k and $23k a year, depending on whether they're part time or full-time.
So yeah...I could survive here on minimum wage, and most of the local stores and restaurants will actually net you more.