Periodic Reporting for period 4 - AUTAR (A Unified Theory of Algorithmic Relaxations)
Reporting period: 2019-12-01 to 2020-09-30
"One of the declared objectives of the project proposal was to push
forward the a priori observation that two methods from different areas
for designing algorithms for constraint satisfaction problems (CSPs)
match in strength: 1) the hierarchies of relaxations arising from the
theory of mathematical programming, and 2) the hierarchies of
relaxations arising from the theory of definability in mathematical
logic. To what extent can this matching in strength be considered a
coincidence or the tip of a deeper unified theory of algorithmic
relaxations? The importance of building a unified theory of
algorithmic relaxations stems from the general principle that, in
applied mathematics, computer science, and the sciences at large,
unified theories tend to have much deeper predicting and explanation
power than a separate collection of unrelated looking techniques.
A second declared objective of the project proposal was to contribute
to the development of an emerging theory of so-called ""satisfiability
solvers"", a logic-inspired collection of algorithms for solving a
particular type of CSPs. Around the turn of the century, a new
algorithmic method to explore hugely vast solution spaces of
combinatorial problems was invented. The excitment that followed the
new discovery launched a race to produce faster and faster computer
programs for solving the notoriously hard satisfiability problem: the
SAT-solvers. The new technology became robust and applications were
found in many areas of industry and of pure mathematics and computer
science themselves. However, despite the great initial progress, the
question of establishing the theoretical foundations that would
explain such great success remained ellusive. In the years around the
beginning of the AUTAR project, the question caught the attention of
computer science theoreticians, and here is where AUTAR comes in. One
of the declared goals of the project was to develop some further
understanding of the theory of SAT-solving with the hope that it would
lead to new ideas that could trigger a new boost."
forward the a priori observation that two methods from different areas
for designing algorithms for constraint satisfaction problems (CSPs)
match in strength: 1) the hierarchies of relaxations arising from the
theory of mathematical programming, and 2) the hierarchies of
relaxations arising from the theory of definability in mathematical
logic. To what extent can this matching in strength be considered a
coincidence or the tip of a deeper unified theory of algorithmic
relaxations? The importance of building a unified theory of
algorithmic relaxations stems from the general principle that, in
applied mathematics, computer science, and the sciences at large,
unified theories tend to have much deeper predicting and explanation
power than a separate collection of unrelated looking techniques.
A second declared objective of the project proposal was to contribute
to the development of an emerging theory of so-called ""satisfiability
solvers"", a logic-inspired collection of algorithms for solving a
particular type of CSPs. Around the turn of the century, a new
algorithmic method to explore hugely vast solution spaces of
combinatorial problems was invented. The excitment that followed the
new discovery launched a race to produce faster and faster computer
programs for solving the notoriously hard satisfiability problem: the
SAT-solvers. The new technology became robust and applications were
found in many areas of industry and of pure mathematics and computer
science themselves. However, despite the great initial progress, the
question of establishing the theoretical foundations that would
explain such great success remained ellusive. In the years around the
beginning of the AUTAR project, the question caught the attention of
computer science theoreticians, and here is where AUTAR comes in. One
of the declared goals of the project was to develop some further
understanding of the theory of SAT-solving with the hope that it would
lead to new ideas that could trigger a new boost."
"The main achievement of the AUTAR project came totally unexpected
while studying and developing the theoretical foundations of
SAT-solving. The goal of that study was to contribute towards a theory
to better understand the proof-search algorithms that underly the
SAT-solving technology. A question that was thought very difficult at
the time of writing the project proposal was whether the proof-search
problem for Resolution, the proof system that underlies most
SAT-solvers, could be solved in polynomial time. Some evidence against
it was known, but no definite answer was in sight. The main result of
the AUTAR project is, contrary to expectations at the time of writing
the proposal, a complete solution to this problem. In the publication
titled ""Automating Resolution is NP-Hard"" we proved that the
proof-search problem for Resolution cannot be solved in polynomial
time, even approximately, unless P = NP. What this means is that,
contrary to empirical evidence, the outstanding performance of
SAT-solvers cannot be completely explained by their ability to find
short Resolution proofs when they exist. Independently of its
theoretical value, whether this discovery can lead to new ideas that
could improve the empirical performance of SAT-solvers even further
remains open for further study.
The solution to the automatiability problem for Resolution resolved a
well-known and long-standing problem that had remained open since at
least 2001 when the last partial progress on the problem was made. The
conference version of the publication was the recipient of the best
paper award of the conference FOCS 2019. The final version of the
article was published in the flagship journal for computer science,
the Journal of the ACM. Several follow up works were published since
the result was announced in 2019, extending the result to stronger
proof systems beyond Resolution.
Besides the important and unexpected achievements reported in the
previous two paragraphs, some of the original goals of the project
were also met with success. One of the important achievements of the
AUTAR project is that the very first question that initiated the
research for a unified theory of algorithmic relaxations was
successfully solved. The starting point for the project was the
finding that, for the graph isomorphism problem, the linear
programming hierarchy of relaxations matched in strength with the
number-of-variables hierarchy of relaxations in mathematical
logic. The question we asked concerned the exact strength of the
semidefinite programming hierarchy in this context: could semidefinite
programming techniques be strictly more powerful that linear
programming techniques in distinguishing non-isomorphic graphs? In
several other contexts, such as in approximating solutions to
constraint satisfaction problems, semidefinite programming provably
beats linear programming. Surprisingly, our main finding in this arena
is the fact that, for the graph isomorphism problem, this is not the
case. In technical terms we say that semidefinite programming
""collapses"" to linear programming for the isomorphism problem. These
results were published as ""Definable Ellipsoid Method, Sums-of-Squares
Proofs, and the Isomorphism Problem"", orginally published in the
Proceedings of LICS 2018. In its final form, the work is currently
submitted to a specialized journal and still under revision. This
finding could have impact for the analysis of certain semidefinite
programming based algorithms in the context of constraint satisfaction
problems, where unlike the linear programming counterparts, their exact
strength is not yet well understood."
while studying and developing the theoretical foundations of
SAT-solving. The goal of that study was to contribute towards a theory
to better understand the proof-search algorithms that underly the
SAT-solving technology. A question that was thought very difficult at
the time of writing the project proposal was whether the proof-search
problem for Resolution, the proof system that underlies most
SAT-solvers, could be solved in polynomial time. Some evidence against
it was known, but no definite answer was in sight. The main result of
the AUTAR project is, contrary to expectations at the time of writing
the proposal, a complete solution to this problem. In the publication
titled ""Automating Resolution is NP-Hard"" we proved that the
proof-search problem for Resolution cannot be solved in polynomial
time, even approximately, unless P = NP. What this means is that,
contrary to empirical evidence, the outstanding performance of
SAT-solvers cannot be completely explained by their ability to find
short Resolution proofs when they exist. Independently of its
theoretical value, whether this discovery can lead to new ideas that
could improve the empirical performance of SAT-solvers even further
remains open for further study.
The solution to the automatiability problem for Resolution resolved a
well-known and long-standing problem that had remained open since at
least 2001 when the last partial progress on the problem was made. The
conference version of the publication was the recipient of the best
paper award of the conference FOCS 2019. The final version of the
article was published in the flagship journal for computer science,
the Journal of the ACM. Several follow up works were published since
the result was announced in 2019, extending the result to stronger
proof systems beyond Resolution.
Besides the important and unexpected achievements reported in the
previous two paragraphs, some of the original goals of the project
were also met with success. One of the important achievements of the
AUTAR project is that the very first question that initiated the
research for a unified theory of algorithmic relaxations was
successfully solved. The starting point for the project was the
finding that, for the graph isomorphism problem, the linear
programming hierarchy of relaxations matched in strength with the
number-of-variables hierarchy of relaxations in mathematical
logic. The question we asked concerned the exact strength of the
semidefinite programming hierarchy in this context: could semidefinite
programming techniques be strictly more powerful that linear
programming techniques in distinguishing non-isomorphic graphs? In
several other contexts, such as in approximating solutions to
constraint satisfaction problems, semidefinite programming provably
beats linear programming. Surprisingly, our main finding in this arena
is the fact that, for the graph isomorphism problem, this is not the
case. In technical terms we say that semidefinite programming
""collapses"" to linear programming for the isomorphism problem. These
results were published as ""Definable Ellipsoid Method, Sums-of-Squares
Proofs, and the Isomorphism Problem"", orginally published in the
Proceedings of LICS 2018. In its final form, the work is currently
submitted to a specialized journal and still under revision. This
finding could have impact for the analysis of certain semidefinite
programming based algorithms in the context of constraint satisfaction
problems, where unlike the linear programming counterparts, their exact
strength is not yet well understood."
The originating idea for the project was that of employing an unconventional
approach to attack otherwise conventional problems. For instance, for
the question of comparing the power of semidefinite versus linear
programming relaxations for the graph isomorphism problem, the
proposal to attack it by methods of mathematical logic is
unconventional. While the outcome of our investigation is a complete
answer to this question that can be stated without any reference to
logic, the path towards a proof required the use of logic methods. It
is in this sense that our approach is unconventional. At the same
time, our approach took us beyond the state of the art as it
established a structural result that was not known to hold
before. Since the results of the publication that led to the answer of
this question are even more general, it is likely that they may
produce further surprises in the future.
approach to attack otherwise conventional problems. For instance, for
the question of comparing the power of semidefinite versus linear
programming relaxations for the graph isomorphism problem, the
proposal to attack it by methods of mathematical logic is
unconventional. While the outcome of our investigation is a complete
answer to this question that can be stated without any reference to
logic, the path towards a proof required the use of logic methods. It
is in this sense that our approach is unconventional. At the same
time, our approach took us beyond the state of the art as it
established a structural result that was not known to hold
before. Since the results of the publication that led to the answer of
this question are even more general, it is likely that they may
produce further surprises in the future.