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**11 - 12**of**12**### Search versus Decision for Election Manipulation Problems ∗

"... Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for man ..."

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Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If integer factoring is hard, then for election manipulation, election bribery, and some types of election control, there are election systems for which recognizing which instances can be successfully manipulated is in polynomial time but producing the successful manipulations cannot be done in polynomial time.

### On Computing the Smallest Four-Coloring of Planar Graphs and Non-Self-Reducible Sets in P

, 2006

"... We show that computing the lexicographically first four-coloring for planar graphs is ∆ p 2-hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P � = NP. We discuss thi ..."

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We show that computing the lexicographically first four-coloring for planar graphs is ∆ p 2-hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P � = NP. We discuss this application to non-self-reducibility and provide a general related result. We also discuss when raising a problem’s NP-hardness lower bound to ∆ p 2-hardness can be valuable.