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The complexity of central slice functions

1986
*
Theoretical Computer Science
*

large

doi:10.1016/0304-3975(86)90122-2
fatcat:rjbhxuvatzbitph77th2733yvi
*combinational**complexity*, then*some*slice off must have large*monotone**complexity*. ... Berkowitz has shown that sufficiently large superlinear lower bounds on the*monotone*network*complexity**of*fk imply lower bounds*of*the same order on the*combinational**complexity**of*f, and that iff has ... Unfortunately, none*of*these results imply superlinear lower bounds on the*combinational**complexity**of*any*function*or set*of**functions*since no efficienl simulation*of**combinational*networks by*monotone*...##
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The complexity of negation-limited networks — A brief survey
[chapter]

1975
*
Lecture Notes in Computer Science
*

*combinational*

*complexity*. ... Unfortunately, few techniques exist for proving lower hounds on the

*combinational*

*complexity*

*of*specific

*functions*

*of*interest, even though the following theorem shows that "most"

*functions*are hard. 2 ...

##
###
On monotone simulations of nonmonotone networks

1989
*
Theoretical Computer Science
*

= the set

doi:10.1016/0304-3975(89)90142-4
fatcat:duhyw4abxrb3rmao2ggv2tio5q
*of*all n-argument*monotone*Boolean*functions*, fk (X,) = k-slice*of**some**function*f (X,,) in Mn 9 c r a = i = 1 ai, C(f) =*combinational**complexity**of*J; C"(f) =*monotone*network*complexity*... Recent work*of*Berkowitz [3] , Wegener [is, 151 and Dunne [S, 6) has considered the problem*of*relating the*combinational*and*monotone*network*complexity**of**monotone*Boolean*functions*via slice*functions*... The result*of*Section 3 shows that the last relation*of*Corollary 5 is in fact the best possible, since there it was proved that C"*(J) = O(&'/log n) for any*function*in this final class. References ...##
###
The gap between monotone and non-monotone circuit complexity is exponential

1988
*
Combinatorica
*

Razborov has shown that there exists a polynomial time computable

doi:10.1007/bf02122563
fatcat:vkdl2vzfifdjpjz7b5knnp57am
*monotone*Booleanfunction whose*monotone*circuit*complexity*is at least n clos ". ... The proof is immediate by*combining*the Alon-Boppana version*of*another argument*of*Razborov with results*of*Grotschel-Lovasz-Schrijver on the Lovasz -capacity, 9*of*a graph. ... I would like to thank Laszlo Babai for many helpful discussions, and Laszlo Lovasz for pointing out an error in an earlier version*of*this note. ...##
###
On the complexity of slice functions

1985
*
Theoretical Computer Science
*

By a result

doi:10.1016/0304-3975(85)90209-9
fatcat:zmdf2oxlcngf5gg4htxa4s3qui
*of*Berkowitz (1982) , the*monotone*circuit*complexity**of*slice*functions*cannot be much larger than the circuit (*combinational*)*complexity**of*these*functions*for arbitrary complete bases. ... All the main methods known for proving lower bounds on the*monotone**complexity**of*Boolean*functions*fail to work in their present form for slice*functions*. ... For the computation*of*Boolean*functions*we consider Boolean circuits (for the definition and elementary properties, see [6] ) and the circuit*complexity*(*combinational**complexity*) either over the complete ...##
###
On the Negation-Limited Circuit Complexity of Merging
[chapter]

1999
*
Lecture Notes in Computer Science
*

A negation-limited circuit is a

doi:10.1007/3-540-48686-0_20
fatcat:ub6elrmuofac7k4vu32d7r2x2u
*combinational*circuit that consists*of*AND, OR gates and a limited number*of*NOT gates. In this paper, we investigate the*complexity**of*negation-limited circuits. ... We prove that the size*complexity**of*the (n; n) merging*function*with t = (log 2 log 2 n − a) NOT gates is (2 a n). ? ...*Combining*this and the fact that the*monotone*circuit*complexity**of*MERGE(n=2 t ; n=2 t ) is ((n=2 t )log(n=2 t )) = (2 a n) to obtain the lower bound for the size*of*C. ...##
###
On the negation-limited circuit complexity of merging

2003
*
Discrete Applied Mathematics
*

A negation-limited circuit is a

doi:10.1016/s0166-218x(02)00215-9
fatcat:hgx6dakwazbrvjhqopu3ohqauy
*combinational*circuit that consists*of*AND, OR gates and a limited number*of*NOT gates. In this paper, we investigate the*complexity**of*negation-limited circuits. ... We prove that the size*complexity**of*the (n; n) merging*function*with t = (log 2 log 2 n − a) NOT gates is (2 a n). ? ...*Combining*this and the fact that the*monotone*circuit*complexity**of*MERGE(n=2 t ; n=2 t ) is ((n=2 t )log(n=2 t )) = (2 a n) to obtain the lower bound for the size*of*C. ...##
###
Fast submodular maximization subject to k-extendible system constraints
[article]

2018
*
arXiv
*
pre-print

submodular

arXiv:1811.07673v1
fatcat:ib3virxnsjgkvozfwvylqlum6q
*functions*and*of*(p(1-p)-ϵ) for non-*monotone*cases with expected computational*complexity**of*only O(pn/ϵr/ϵ), where r is the largest size*of*the feasible solutions, 0<p ≤1/1+k is the sampling ...*Combining*the sampling process and the decreasing threshold strategy, our algorithm Sample Decreasing Threshold Greedy Algorithm (SDTGA) obtains an expected approximation guarantee*of*(p-ϵ) for*monotone*... For non-*monotone*submodular*functions*, Gupta et al. [25] provided an approximation ratio*of*k (k+1)(3k+3) with time*complexity**of*O(nrk). ...##
###
Lower Bounds for Testing Properties of Functions over Hypergrid Domains

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

We use this method to prove strong, and in many cases optimal, lower bounds on the query

doi:10.1109/ccc.2014.38
dblp:conf/coco/BlaisRY14
fatcat:dj4ouaxx25akhbinvxpnfo7hoi
*complexity**of*testing fundamental properties*of**functions*f : {1, . . . , n} d → R over hypergrid domains:*monotonicity*... We show how the communication*complexity*method introduced in (Blais, Brody, Matulef 2012) can be used to prove lower bounds on the number*of*queries required to test properties*of**functions*with non-hypercube ... We also thank Joshua Brody and Oded Goldreich for insightful conversations about the communication*complexity*method. ...##
###
Combining testing and proof to gain high assurance in software: A case study

2013
*
2013 IEEE 24th International Symposium on Software Reliability Engineering (ISSRE)
*

This approach

doi:10.1109/issre.2013.6698924
dblp:conf/issre/BishopBC13
fatcat:qwnfbtig3jdurggzria3vhn7b4
*combines*a*monotonicity*analysis with a defined set*of*tests to demonstrate the accuracy*of*a software*function*over its entire input range. ... This paper presents a specific example*of*this approach applied to the verification*of*continuous*monotonic**functions*. ... The*combination**of*static and dynamic analysis comprises: • a formal proof*of**some*property P*of*program f(⋅) • a specific set*of*test inputs T drawn from the input domain I that meet*some*success criterion ...##
###
Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

2020
*
Algorithmica
*

Euler characteristic

doi:10.1007/s00453-020-00676-9
pmid:32801408
pmcid:PMC7416760
fatcat:s7i4a6yrtjd43okvw3r4cmreqe
*of*the associated simplicial (graph)*complex**of*Φ is non-zero. ...*monotone*property Φ that does not hold on the Hamilton cycle as well as for*some**monotone*properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for k > 2 . ... Furthermore we wish to point out that Sloane's OEIS [1] was very useful in the course*of*this work. ...##
###
Page 5428 of Mathematical Reviews Vol. , Issue 82m
[page]

1982
*
Mathematical Reviews
*

The algorithm makes use

*of*the Boolean difference technique.” Tosi¢, Ratko 82m:94062 An optimal identification algorithm for*some*subclasses*of**monotone*Boolean*functions*. ... The author assumes that Boolean*functions*belong to*some*class*of**monotone*ones and gives an identification procedure. W. Greblicki (Wroclaw) Wu, X.; Hurst, S. ...##
###
Page 2651 of Mathematical Reviews Vol. , Issue 88e
[page]

1988
*
Mathematical Reviews
*

*complexity*

*of*f implies large

*monotone*

*complexity*

*of*

*some*k-slice

*of*f which is not effectively specified. ... Further, it has been shown that large lower bounds on the

*monotone*

*complexity*

*of*

*some*k-slice

*of*f imply lower bounds

*of*the same order on the com- binational

*complexity*

*of*{, and conversely, large

*combinational*...

##
###
Lower bounds for monotone span programs

1995
*
Proceedings of IEEE 36th Annual Foundations of Computer Science SFCS-95
*

The main result proved here yields quadratic lower bounds for the size

doi:10.1109/sfcs.1995.492669
dblp:conf/focs/BeimelGP95
fatcat:vc4la34uarh5ji53esx6wsueya
*of**monotone*span programs, improving on the largest previously known bounds for explicit*functions*. ... The bound is asymptotically tight for the*function*corresponding to a class*of*4-cliques. ... Next we present a*monotone*span program*of*linear size (exactly m) for a*function*on m variables, that is known to have ((m= log m) 3=2 )*monotone*circuit*complexity*. ...##
###
Page 6983 of Mathematical Reviews Vol. , Issue 96k
[page]

1996
*
Mathematical Reviews
*

Since for slice

*functions*the*combinational**complexity*and the*monotone**complexity*are closely related, it was assumed that the investiga- tion*of*the*monotone**complexity**of*slice*functions*would yield ... an important approach for proving nonlinear lower bounds on the*combinational**complexity**of*Boolean*functions*. ...
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