Abstract
A longstanding open question is whether there is an equivalence between the computational task of determining the minimum size of any circuit computing a given function and the task of producing a minimumsized circuit for a given function. While it is widely conjectured that both tasks require "perebor," or bruteforce search, researchers have not yet ruled out the possibility that the search problem requires exponential time but the decision problem has a linear time algorithm.
In this paper, we make progress in connecting the search and decision complexity of minimizing formulas. Let MFSP denote the problem that takes as input the truth table of a Boolean function f and an integer size parameter s and decides whether there is a formula for f of size at most s. Let Search denote the corresponding search problem where one has to output some optimal formula for computing f.
Our main result is that given an oracle to MFSP, one can solve SearchMFSP in time polynomial in the length N of the truth table of f and the number t of "nearoptimal" formulas for f, in particular O(N⁶t²)time. While the quantity t is not well understood, we use this result (and some extensions) to prove that given an oracle to MFSP:
 there is a deterministic 2^O(N/(log log N))time oracle algorithm for solving SearchMFSP on all but a o(1)fraction of instances, and
 there is a randomized O(2^.67N)time oracle algorithm for solving SearchMFSP on all instances. Intriguingly, the main idea behind our algorithms is in some sense a "reverse application" of the gate elimination technique.
BibTeX  Entry
@InProceedings{ilango:LIPIcs:2020:12583,
author = {Rahul Ilango},
title = {{Connecting Perebor Conjectures: Towards a Search to Decision Reduction for Minimizing Formulas}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {31:131:35},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771566},
ISSN = {18688969},
year = {2020},
volume = {169},
editor = {Shubhangi Saraf},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12583},
URN = {urn:nbn:de:0030drops125834},
doi = {10.4230/LIPIcs.CCC.2020.31},
annote = {Keywords: minimum circuit size problem, minimum formula size problem, gate elimination, search to decision reduction, selfreducibility}
}
Keywords: 

minimum circuit size problem, minimum formula size problem, gate elimination, search to decision reduction, selfreducibility 
Collection: 

35th Computational Complexity Conference (CCC 2020) 
Issue Date: 

2020 
Date of publication: 

17.07.2020 