Truth vs Proof: The Unique Games Conjecture and Feige’s Hypothesis

Theoretical Computer Science is blessed (or cursed?) with many open problems. For some of these questions, such as the $latex {P}&fg=000000$ vs $latex {NP}&fg=000000$ problem, it seems like it could be decades or more before they reach resolution. So, if we have no proof either way, what do we assume about the answer? We could … Continue reading Truth vs Proof: The Unique Games Conjecture and Feige’s Hypothesis

The Switching Lemma

Hastad's Switching Lemma is one of the gems of computational complexity. The switching lemma analyzes the effect of random restrictions on $latex AC^0$ circuits, which are constant depth circuits with AND, OR and NOT gates of unbounded fanin. It says that such restrictions tends to simplify  $latex AC^0$ circuits in surprising ways. An immediate consequence of the switching lemma is strong … Continue reading The Switching Lemma

Higher Lower Bounds: Just Ask for More!

In memory of Mihai Pătraşcu. Continuing the spirit of the previous post, in this post I will describe a specific technique for proving lower bounds for (static) data structures, when the space is (near) linear. This technique was introduced in the paper by Mihai and Mikkel Thorup, called "HIGHER lower Bounds for Near-Neighbor and  F … Continue reading Higher Lower Bounds: Just Ask for More!

The Art of Reductions

In memory of Mihai  Pătraşcu Written by Rasmus Pagh, Rina Panigrahy, Kunal Talwar and Udi Wieder Reductions are arguably at the heart of complexity theory. They show that one problem is at least as hard as another. Finding reductions often requires creativity and considerable technical skill; some reductions seem as if they were pulled from thin air. The most difficult … Continue reading The Art of Reductions

“Just a Spoonful of Sugar …”

Tim Roughgarden sent me the following email. I found this idea so refreshing that I thought I should share more widely. Well done PC! -------------------------------------------------------------------------------- PC meetings are typically all work and no play.  But the FOCS '12 PC, in an act of rebellion, has decided to spend a day disucssing their own results, before … Continue reading “Just a Spoonful of Sugar …”