Define single source shortest path algorithm
As expected, this hardness can be extended to larger values of $\chi_d$ for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any $\chi_d \ge 2$, and hence 2-coloring is the only hard case for this parameter.In addition to the above, we give an ETH-based lower bound for treewidth and pathwidth, showing that no algorithm can solve the problem in $n^$, essentially matching the complexity of an algorithm obtained with standard techniques.We give a deterministic algorithm which, given an $n$-variable $\mathrm(n)$-clause CNF formula $F$ that has at least $\varepsilon 2^n$ satisfying assignments, runs in time \[ n^ \] for $\varepsilon \ge 1/\mathrm(n)$ and outputs a satisfying assignment of $F$.Prior to our work the fastest known algorithm for this problem was simply to enumerate over all seeds of a pseudorandom generator for CNFs; using the best known PRGs for CNFs [DETT10], this takes time $n^$ even for constant $\varepsilon$.One could say that if the author’s identity is revealed then there is no harm, since in such a case we simply revert to the original form of non anonymous submissions.However, the fact that the authors’ identity is known to participants in the process (e.g., maybe some reviewers but not others), makes some conflicts and biases invisible.Nor do I have any principled objection to anonymization: I do for example practice anonymous grading in my courses for exactly this reason.I also don’t buy the suggestion that we must know the author’s identity to evaluate if the proof is correct.
We also give a $(tw)^$ algorithm which achieves the desired $\Delta^*$ exactly while 2-approximating the minimum value of $\chi_d$.Moreover, the fact that the author’s identity is not “officially” known, causes a lot of practical headaches.For example, as a PC member you can’t just shoot a quick email to an expert to ask for a quick opinion on the paper, since they may well be the author themselves (as happened to me several time as a CRYPTO PC member), or someone closely related to them.We show that this is close to optimal, by establishing that no FPT algorithm can (under standard assumptions) achieve a better than /2$-approximation to $\chi_d$, even when an extra constant additive error is also allowed. Servedio, Li-Yang Tan Download: PDFAbstract: We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\varepsilon$-PRG for the class of size-$M$ depth-$d$ $\mathsf^0$ circuits with seed length $\log(M)^\cdot \log(1/\varepsilon)$, and an $\varepsilon$-PRG for the class of $S$-sparse $\mathbb_2$ polynomials with seed length ^\cdot \log(1/\varepsilon)$.These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds.
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In the latter case, and in particular in theoretical computer science, you often need the expertise of very particular people that have worked on this area.