Nullstellensatz Certificates for NP-complete Problems

The core concept is representing combinatorial decision problems using polynomial equations. A problem's feasibility is directly linked to the existence of a solution within a related polynomial system over an algebraically-closed field K. Hilbert's Nullstellensatz provides a certificate of infeasibility when no solution exists. The document focuses on leveraging this algebraic framework to create efficient algorithms for solving combinatorial problems, particularly those involving graphs. The use of polynomial system encodings for solving combinatorial problems, while theoretically sound, hasn't seen widespread computational application. The document aims to investigate using these systems to efficiently determine if a graph, or similar structure, possesses a specific property defined by the polynomial system. This is termed the 'combinatorial feasibility problem,' and the authors are particularly interested in the practical applicability to large structures, such as graphs with many vertices. Existing methods such as Gröbner bases are known to be effective but limited by computational constraints, especially when dealing with hundreds of equations. This is why the Nullstellensatz Linear Algebra algorithm (NulLA) is introduced as an alternative.

The document provides concrete encodings for various NP-complete and other combinatorial problems. These include the independent set problem, graph k-colorability (with a specific focus on graph 3-colorability), the Hamiltonian cycle problem, graph planarity, edge-chromatic number, max-cut, and the Boolean satisfiability problem (SAT). The encodings are presented not just to illustrate diverse encoding techniques, but also to allow for a computational comparison between them in later sections. The encoding for SAT is particularly important because polynomial reductions to SAT can be used to encode all NP-complete problems. However, the document notes that this approach can be computationally impractical due to an increase in the number of variables and equations. The effectiveness of different types of constraints is discussed. For instance, encodings employing constraints of the form xᵢ(xᵢ − 1) are noted to perform poorly with algebraic methods like NulLA, whereas roots of unity constraints show better practical performance. The bijective mapping between solutions and combinatorial structures (such as independent sets and k-colorings) is rigorously proven for several encodings, solidifying the validity of the polynomial representations.

The document emphasizes that multiple encodings for the same problem often exist, and the choice of encoding significantly impacts the computational efficiency. The goal is to find encodings that are 'algebraically compatible' with Hilbert's Nullstellensatz, leading to computationally tractable certificates. The size and structure of the resulting polynomial systems are crucial for computational feasibility. While the document highlights the theoretical possibility of encoding any combinatorial problem as a polynomial system, it underscores the practical challenges of managing the complexity of those encodings, especially for large problems. This includes challenges with encoding size; for instance, the Hamiltonian cycle problem encoding can exhibit exponential growth in size for certain graphs. Despite this, the document demonstrates the possibility of using polynomial representations effectively, particularly for k-regular graphs where more concise encodings are possible. The chapter concludes by presenting encodings for a range of problems, setting the stage for further analysis and comparison of their computational behavior.

The Nullstellensatz Linear Algebra algorithm (NulLA) is designed to determine the feasibility of combinatorial problems encoded as polynomial systems. Its core functionality relies on Hilbert's Nullstellensatz, specifically exploiting the experimentally observed low degree of the associated Nullstellensatz certificates. The algorithm's efficiency hinges on this low degree and the resulting manageable size of the sparse linear algebra computations involved. NulLA works iteratively, solving a series of linear algebra systems of increasing degree. A feasible system indicates infeasibility in the original combinatorial problem, providing a certificate proving this infeasibility. The iterative nature of the algorithm guarantees termination because known upper bounds exist for the degrees of the polynomials within the Nullstellensatz certificates (as referenced in [30]). While general upper bounds from commutative algebra are available, the algorithm's practical success depends on finding tighter bounds specific to the combinatorial ideals being considered. The overall runtime is directly determined by the number of iterations needed to either find a solution (and hence, a certificate) or to exhaust all possible degrees.

NulLA's central idea is to generate a finite sequence of linear algebra systems based on Nullstellensatz certificates of progressively higher degrees. The algorithm determines whether a certificate of a given degree exists by solving a corresponding system of linear equations. The variables in this linear system directly map to the coefficients of the monomials within the coefficient polynomials (β₁, ..., βₛ) of the certificate. Finding a solution to this linear system means a certificate has been found, thus proving the infeasibility of the underlying combinatorial problem. If no solution exists at a given degree, the algorithm iterates to a higher degree. This process is guaranteed to eventually terminate because there are theoretical bounds (though often very large) on the maximum possible degree of these coefficient polynomials that must hold if a Nullstellensatz certificate exists. The algorithm's efficiency is therefore strongly tied to the actual degree of the minimal certificate for the problem instance, as this directly governs the size of the linear system that needs to be solved. The chapter concludes with mention of further efficiency gains which can be made, including utilizing finite field calculations and exploiting symmetry to reduce matrix size.

The document acknowledges that even low-degree Nullstellensatz certificates can result in substantial linear systems in practice, posing a significant computational challenge. The authors explore several strategies to mitigate this. These include techniques to reduce the degree of the certificates (and thus the size of the resulting linear systems), as well as methods for eliminating monomials from certificates, particularly for graph k-colorability problems. A probabilistic approach is also described that reduces the number of unknowns in the linear systems. While this probabilistic method shows promise, achieving a 90% success rate with a 60% reduction in variables, the results are less favorable for graph 3-colorability over the field F₂. The document notes a significant computational limitation—for some very large graphs (example: C4000.5.clq), the size of the linear systems becomes incredibly large, making computation infeasible even with degree one certificates. This highlights the need for further optimization of the underlying linear algebra techniques. The algorithm's efficiency is also compared to other methods such as Gröbner bases and the Alon-Tarsi method, demonstrating advantages in terms of scalability but also highlighting that certain special cases can make other methods considerably faster.

This section delves into the relationship between the complexity of Nullstellensatz certificates and the underlying computational complexity of the associated combinatorial problems. A key focus is on the certificate's degree and the number of monomials it contains. The text explores the implications of assuming P ≠ NP and NP ≠ coNP. Under this assumption, it's argued that there must exist infinite families of problem instances where the minimum degree of the associated Nullstellensatz certificate grows linearly with the input size, and the certificates themselves are super-polynomially dense in terms of the number of monomials. Specific examples are given: for graph 3-colorability, the minimum degree of the certificate follows the sequence 1, 4, 7, ...; for the independent set problem, the minimum degree is shown to be equal to the size of the graph's largest independent set, with one monomial for each independent set in the graph. These findings highlight a connection between the certificate's structural complexity and the inherent hardness of the underlying combinatorial problem.

The section shifts its focus from finding Nullstellensatz certificates to verifying them, linking this process to the relationship between NP and coNP. The core argument centers on the fact that Nullstellensatz certificates are polynomial identities that simplify to one. Thus, verifying a certificate involves evaluating a polynomial expression. The complexity of this evaluation is crucial. It's argued that not only must the degree of the certificate grow with input size (as discussed previously), but also the number of operations required to simplify the expression must be large, implying a dense structure for the certificates. This discussion relates to the inherent difficulty of verifying 'no' instances of NP-complete problems. If 'no' instances can be verified quickly, then it means NP=coNP which contradicts the initial assumption of P≠NP and NP≠coNP. This section sets the stage for investigating the use of probabilistic methods for efficient verification.

To address the potential computational bottleneck of verifying large, dense Nullstellensatz certificates, the section introduces the Schwartz-Zippel lemma. This lemma provides a probabilistic method for verifying polynomial identities. Instead of fully expanding and simplifying the certificate (a potentially exponential-time process), the lemma allows for a probabilistic verification that can be performed in polynomial time. The Schwartz-Zippel algorithm involves randomly evaluating the polynomial at several points. If the polynomial evaluates to zero at all tested points, then it's probabilistically determined to be identically zero. The probability of error can be controlled by adjusting the number of test points and the size of the field from which the random points are selected. This approach offers a significant potential speedup for verifying Nullstellensatz certificates, particularly when deterministic methods become computationally prohibitive. The section concludes by highlighting this as a potential path towards efficient verification, especially given that a Nullstellensatz certificate is, in essence, a polynomial that simplifies to one (and therefore, subtracting one from the certificate yields an identically zero polynomial).

This section focuses on the structure and properties of Nullstellensatz certificates specifically for the independent set problem, using the encoding described by Lovász. A key result is the direct relationship established between these certificates and the underlying graph's properties. The minimum degree of the certificate is proven to be equal to the stability number α(G), representing the size of the largest independent set within the graph. Furthermore, a one-to-one correspondence is shown between the monomials in the certificate and the independent sets in the graph. This remarkable connection provides a clear and unexpected link between the algebraic structure of the certificate and the graph's combinatorial properties. Importantly, this analysis is conducted without relying on assumptions about complexity classes (such as P vs. NP or NP vs. coNP). The result directly answers an open question posed by Lovász [41] concerning the minimum degree of Nullstellensatz certificates in specific graph families. The findings demonstrate not only linear growth in the minimum degree of the certificates but also that they are dense, meaning they contain a significant number of monomials. This density is a crucial aspect that relates to the computational cost of certificate verification.

The results on the independent set problem's Nullstellensatz certificates have significant implications for complexity theory. The demonstrated linear growth of the minimum degree and the density of the certificates are consistent with the predictions made by assuming P ≠ NP and NP ≠ coNP. However, crucially, these findings are obtained without making any assumptions about the relationship between these complexity classes. This direct, unconditional proof of linear growth and density in the certificates provides a concrete answer to an open question posed by Lovász [41] which challenged researchers to find explicit families of graphs exhibiting such growth in the minimum degree of their Nullstellensatz certificates. This work contributes to our understanding of the connection between the algebraic complexity represented by Nullstellensatz certificates and the inherent computational complexity of the underlying combinatorial problems. The work demonstrates that the certificates themselves contain all the information about the 'yes' instances of the problem, effectively encoding all possible solutions within the structure of the 'no' instance's certificate.

A key aspect of the analysis is the proof of the density of the Nullstellensatz certificates for the independent set problem. This density, coupled with the linear growth in the minimum degree, signifies a significant computational challenge. The work establishes new lower bounds for both the degree and the number of terms in Nullstellensatz certificates, contributing to the existing research on this topic. Previous work had established both logarithmic and linear growth in specific cases or over finite fields; however, this work offers a more robust and general result. The fact that the certificate for a 'no' instance inherently encapsulates all the information regarding 'yes' instances is highlighted as a noteworthy finding. The section concludes by emphasizing the surprising clarity with which Hilbert's Nullstellensatz captures combinatorial properties of the graph, highlighting the potential for deeper investigation into this unexpected connection between algebraic structures and combinatorial problems. The explicit relationship between the certificate and the graph's independent sets could potentially lead to new insights into algorithms and complexity.

This section investigates the minimum degree of Nullstellensatz certificates for non-3-colorable graphs. A key result establishes that the minimum degree follows the sequence 1, 4, 7, and so on. This sequence is shown to hold regardless of whether the encoding is done over the complex numbers (C) or the finite field F₂, although the specific minimum degree may differ between the two fields (at least 4 for C and at least 1 for F₂). The study also explores the impact of different encodings on the certificate's degree, indicating that the choice of encoding significantly influences computational complexity. The analysis differentiates between the minimum degree for various encodings, highlighting that certain encodings (such as root of unity constraints) are more computationally favorable than others (such as those using constraints of the form xᵢ(xᵢ − 1)). The section explores the computational implications of these varying minimum degrees, with a clear indication that a lower minimum degree contributes significantly to the efficiency of the algorithm. This sets the stage for the exploration of strategies to reduce the degree of the certificates, thereby optimizing the algorithm's performance.

A significant part of the analysis focuses on the effect of adding extra equations, specifically triangle equations, to the polynomial system. These 'degree-cutter' equations, derived from the presence of triangles within the graph, are shown to reduce the overall degree of the Nullstellensatz certificate needed to prove non-3-colorability. This reduction is demonstrated through an example, showing that for the Koester graph, adding 25 triangle equations reduces the certificate degree from four to one, resulting in a substantial decrease in computation time. The inclusion of these triangle equations is framed as a method to exploit the underlying combinatorial structure of the graph to improve the algorithm's efficiency. The discussion hints at a deeper relationship between the graph's structural properties and the degree of the Nullstellensatz certificates, suggesting avenues for future investigation into graph characteristics that influence computational complexity. The addition of extra equations despite reducing the degree and overall computation time of the certificate is highlighted as a key improvement to the efficiency of the algorithm.

The section presents experimental results investigating the relationship between the degree of the Nullstellensatz certificates and the perceived 'hardness' of non-3-colorable graphs. Two families of graphs known to be computationally challenging for 3-colorability algorithms are examined: Minimum Unsolvable Graphs (MUGs) and 4-critical Graph Units (4-CGUs). Experimental results using NulLA over F₂ show a clear correlation between the degree of the certificates and computational difficulty, with higher-degree certificates corresponding to more computationally expensive instances. The performance of NulLA is also compared with the Gröbner basis method, highlighting both its advantages in scalability and the superior performance of the Gröbner basis method for some of the 'hard' instances. The analysis reveals that for some graphs (such as odd-wheels and Kneser graphs), the degree of the certificates remains low regardless of graph size, indicating families of graphs that do not exhibit degree growth. However, other families (MUGs and 4-CGUs) show a noticeable increase in the degree, highlighting the challenges in predicting and understanding the behavior of the algorithm for increasingly complex graphs. The section concludes by underscoring that identifying and understanding combinatorial properties that are linked to certificate complexity remains an area of active research.

This section presents experimental results comparing the performance of the Nullstellensatz Linear Algebra algorithm (NulLA) against established methods for solving combinatorial problems, specifically focusing on graph 3-colorability. NulLA's performance is compared to Gröbner basis methods and the Alon-Tarsi method. The experimental results indicate that NulLA scales more efficiently with respect to input size than the Gröbner basis method. A key observation is NulLA's favorable comparison with the Alon-Tarsi method, which often fails to terminate within reasonable time limits. However, the Alon-Tarsi method shows significantly faster performance in specific cases where a non-3-colorable subgraph (like a 4-clique) is found early in the graph traversal, due to its iterative implementation. The sensitivity of the Alon-Tarsi method to vertex and edge ordering is emphasized by an example using the ninth Mycielski graph, where the method’s runtime varies drastically depending on the input ordering. This suggests that the incorporation of a similar iterative strategy into NulLA or the Gröbner basis method could enhance their performance in similar special cases. The comparison includes testing on odd-wheel graphs, further highlighting NulLA's comparative efficiency.

The experimental evaluation extends to testing NulLA on computationally challenging instances of graph 3-colorability, specifically using Minimum Unsolvable Graphs (MUGs) from [48] and 4-critical Graph Units (4-CGUs) from [37]. These graph families are known to pose significant computational challenges for existing heuristics and solvers. The experiments involved running NulLA (over F₂) and Gröbner basis methods on these instances. The results indicate that for the MUG family, there's a clear correlation between the increase in graph size and a corresponding increase in the degree of the Nullstellensatz certificates, suggesting that these are 'hard' instances for NulLA. Similarly, the 4-CGU family demonstrated a similar trend in certificate degree growth with increasing graph size. However, the study notes that in some of these cases, the Gröbner basis method outperformed NulLA, potentially suggesting avenues for further algorithm refinement. The lack of degree reduction using triangle equations for these 'hard' instances also highlights the inherent computational difficulties of these problems. The results underscore the need for a deeper understanding of the relationship between graph structure and the properties of the Nullstellensatz certificates to fully grasp the challenges and opportunities associated with these computationally demanding instances.

The concluding section outlines several avenues for future research and algorithm enhancements. Key areas of focus include improving the efficiency of the linear algebra computations within NulLA. The potential of incremental solving strategies is suggested, where the algorithm could reuse computation from previous iterations. The possibility of implementing more advanced linear algebra solvers, such as the block Lanczos algorithm, is discussed as a means of accelerating the solution of the sparse linear systems. Another crucial aspect is the development of a comprehensive library of problem encodings, emphasizing 'algebraic compatibility' with Hilbert's Nullstellensatz. The binary encoding of the independent set problem is presented as an example of an encoding that does not perform well, highlighting the importance of careful encoding design for optimal algorithmic performance. The section also suggests exploring the combinatorial meaning of the certificate coefficients to potentially gain insights into why some instances are computationally harder than others. Finally, the section poses open questions on identifying families of graphs with demonstrably growing certificate degree, and whether these families would present similar challenges to other algorithms.