How do you relax vertex cover to a linear program?

• Delete every second edge from the graph
• Replace each 0 or 1 with a fraction between 0 and 1
• Square every variable to smooth the program

What does a linear program need?

• Variables for choices, linear limits, and a linear goal
• A guess at the answer and nothing else
• Curved limits with a squared goal

The relaxed fractional optimum bounds the integer optimum from below.

A fractional vertex cover optimum costs 6. Rounding at one half at most doubles the cost. What is the worst cost the rounded cover can have?

A dual solution certifies your primal answer is within 5 percent of optimal. What did the dual give you?

• The exact integer optimum itself
• A rounding rule for the fractions
• A certificate for a bound, without knowing the optimum

Why does rounding at one half keep every edge covered?

• Every edge holds two endpoints summing to at least 1, so one rounds up
• Rounding up always deletes the cheaper endpoint
• Halves are lucky numbers for graphs

A student rounds fractions down to 0 to save money and drops some edges. What is the error?

• Rounding down is always optimal for covers
• A cover must cover every edge; uncovered edges break feasibility whatever they save
• Dual certificates forbid all rounding

Primal rounded cost is 11 and a dual solution certifies no answer beats 10. What can you claim?

• The rounded answer is optimal, since 11 is near 10
• The rounded answer is within a factor of 1.1 of optimal
• Nothing, since primal and dual disagree