Weizmann Logo
ECCC
Electronic Colloquium on Computational Complexity

Under the auspices of the Computational Complexity Foundation (CCF)

Login | Register | Classic Style



REPORTS > DETAIL:

Revision(s):

Revision #5 to TR19-106 | 25th October 2021 10:12

Semialgebraic Proofs and Efficient Algorithm Design

RSS-Feed




Revision #5
Authors: Noah Fleming, Pravesh Kothari, Toniann Pitassi
Accepted on: 25th October 2021 10:12
Downloads: 1633
Keywords: 


Abstract:

Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential in the context of semi-algebraic proof systems and basic primitives in algorithm design such as linear and semidefinite programming. The proof system perspective, in this context, has provided fundamentally new tools for both algorithm design and analysis. These news tools have helped in both designing better algorithms for well-studied problems and proving tight lower bounds on such techniques.

This monograph is aimed at expositing this interplay between proof systems and efficient algorithm design and surveying the state-of-the-art for two of the most important semi-algebraic proof systems: Sherali-Adams and Sum-of-Squares.

We rigorously develop and survey the state-of-the-art for Sherali-Adams and Sum-of-Squares both as proof systems, as well as a general family of optimization algorithms, stressing that these perspectives are formal duals to one-another. Our treatment relies on interpreting the outputs of the Sum-of-Squares and Sherali-Adams algorithms as generalized expectation functions -- a viewpoint that has been essential in obtaining both algorithmic results and lower bounds. The emphasis is on illustrating the main ideas by presenting a small fraction of representative results with detailed intuition and commentary. The monograph is self-contained and includes a review of the necessary mathematical background including basic theory of linear and semi-definite programming.



Changes to previous version:

Minor typos fixed


Revision #4 to TR19-106 | 26th December 2020 20:14

Semialgebraic Proofs and Efficient Algorithm Design





Revision #4
Authors: Noah Fleming, Pravesh Kothari, Toniann Pitassi
Accepted on: 26th December 2020 20:14
Downloads: 1204
Keywords: 


Abstract:

Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential in the context of semi-algebraic proof systems and basic primitives in algorithm design such as linear and semidefinite programming. The proof system perspective, in this context, has provided fundamentally new tools for both algorithm design and analysis. These news tools have helped in both designing better algorithms for well-studied problems and proving tight lower bounds on such techniques.

This monograph is aimed at expositing this interplay between proof systems and efficient algorithm design and surveying the state-of-the-art for two of the most important semi-algebraic proof systems: Sherali-Adams and Sum-of-Squares.

We rigorously develop and survey the state-of-the-art for Sherali-Adams and Sum-of-Squares both as proof systems, as well as a general family of optimization algorithms, stressing that these perspectives are formal duals to one-another. Our treatment relies on interpreting the outputs of the Sum-of-Squares and Sherali-Adams algorithms as generalized expectation functions -- a viewpoint that has been essential in obtaining both algorithmic results and lower bounds. The emphasis is on illustrating the main ideas by presenting a small fraction of representative results with detailed intuition and commentary. The monograph is self-contained and includes a review of the necessary mathematical background including basic theory of linear and semi-definite programming.



Changes to previous version:

Minor edits


Revision #3 to TR19-106 | 13th June 2020 07:53

Semialgebraic Proofs and Efficient Algorithm Design





Revision #3
Authors: Noah Fleming, Pravesh Kothari, Toniann Pitassi
Accepted on: 13th June 2020 07:53
Downloads: 1836
Keywords: 


Abstract:

Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential in the context of semi-algebraic proof systems and basic primitives in algorithm design such as linear and semidefinite programming. The proof system perspective, in this context, has provided fundamentally new tools for both algorithm design and analysis. These news tools have helped in both designing better algorithms for well-studied problems and proving tight lower bounds on such techniques.

This monograph is aimed at expositing this interplay between proof systems and efficient algorithm design and surveying the state-of-the-art for two of the most important semi-algebraic proof systems: Sherali-Adams and Sum-of-Squares.

We rigorously develop and survey the state-of-the-art for Sherali-Adams and Sum-of-Squares both as proof systems, as well as a general family of optimization algorithms, stressing that these perspectives are formal duals to one-another. Our treatment relies on interpreting the outputs of the Sum-of-Squares and Sherali-Adams algorithms as generalized expectation functions -- a viewpoint that has been essential in obtaining both algorithmic results and lower bounds. The emphasis is on illustrating the main ideas by presenting a small fraction of representative results with detailed intuition and commentary. The monograph is self-contained and includes a review of the necessary mathematical background including basic theory of linear and semi-definite programming.



Changes to previous version:

Fixed typos and minor corrections throughout.


Revision #2 to TR19-106 | 19th November 2019 17:35

Semialgebraic Proofs and Efficient Algorithm Design





Revision #2
Authors: Noah Fleming, Pravesh Kothari, Toniann Pitassi
Accepted on: 19th November 2019 17:35
Downloads: 1854
Keywords: 


Abstract:

Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential in the context of semi-algebraic proof systems and basic primitives in algorithm design such as linear and semidefinite programming. The proof system perspective, in this context, has provided fundamentally new tools for both algorithm design and analysis. These news tools have helped in both designing better algorithms for well-studied problems and proving tight lower bounds on such techniques.

This monograph is aimed at expositing this interplay between proof systems and efficient algorithm design and surveying the state-of-the-art for two of the most important semi-algebraic proof systems: Sherali-Adams and Sum-of-Squares.

We rigorously develop and survey the state-of-the-art for Sherali-Adams and Sum-of-Squares both as proof systems, as well as a general family of optimization algorithms, stressing that these perspectives are formal duals to one-another. Our treatment relies on interpreting the outputs of the Sum-of-Squares and Sherali-Adams algorithms as generalized expectation functions -- a viewpoint that has been essential in obtaining both algorithmic results and lower bounds. The emphasis is on illustrating the main ideas by presenting a small fraction of representative results with detailed intuition and commentary. The monograph is self-contained and includes a review of the necessary mathematical background including basic theory of linear and semi-definite programming.


Revision #1 to TR19-106 | 19th October 2019 22:22

Semialgebraic Proofs and Efficient Algorithm Design





Revision #1
Authors: Noah Fleming, Pravesh Kothari, Toniann Pitassi
Accepted on: 19th October 2019 22:22
Downloads: 949
Keywords: 


Abstract:

Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential in the context of semi-algebraic proof systems and basic primitives in algorithm design such as linear and semidefinite programming. The proof system perspective, in this context, has provided fundamentally new tools for both algorithm design and analysis. These news tools have helped in both designing better algorithms for well-studied problems and proving tight lower bounds on such techniques.

This monograph is aimed at expositing this interplay between proof systems and efficient algorithm design and surveying the state-of-the-art for two of the most important semi-algebraic proof systems: Sherali-Adams and Sum-of-Squares.

We rigorously develop and survey the state-of-the-art for Sherali-Adams and Sum-of-Squares both as proof systems, as well as a general family of optimization algorithms, stressing that these perspectives are formal duals to one-another. Our treatment relies on interpreting the outputs of the Sum-of-Squares and Sherali-Adams algorithms as generalized expectation functions -- a viewpoint that has been essential in obtaining both algorithmic results and lower bounds. The emphasis is on illustrating the main ideas by presenting a small fraction of representative results with detailed intuition and commentary. The monograph is self-contained and includes a review of the necessary mathematical background including basic theory of linear and semi-definite programming.



Changes to previous version:

-Rewrote introductory section on proof complexity
-editorial changes and notational simplification throughout.


Paper:

TR19-106 | 12th August 2019 20:45

Semialgebraic Proofs and Efficient Algorithm Design


Abstract:

Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential in the context of semi-algebraic proof systems and basic primitives in algorithm design such as linear and semidefinite programming. The proof system perspective, in this context, has provided fundamentally new tools for both algorithm design and analysis. These news tools have helped in both designing better algorithms for well-studied problems and proving tight lower bounds on such techniques.

This monograph is aimed at expositing this interplay between proof systems and efficient algorithm design and surveying the state-of-the-art for two of the most important semi-algebraic proof systems: Sherali-Adams and Sum-of-Squares.

We rigorously develop and survey the state-of-the-art for Sherali-Adams and Sum-of-Squares both as proof systems, as well as a general family of optimization algorithms, stressing that these perspectives are formal duals to one-another. Our treatment relies on interpreting the outputs of the Sum-of-Squares and Sherali-Adams algorithms as generalized expectation functions -- a viewpoint that has been essential in obtaining both algorithmic results and lower bounds. The emphasis is on illustrating the main ideas by presenting a small fraction of representative results with detailed intuition and commentary. The monograph is self-contained and includes a review of the necessary mathematical background including basic theory of linear and semi-definite programming.



ISSN 1433-8092 | Imprint