Algebraic and Combinatorial Methods in Computational Complexity

Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science. The seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress

Derandomizing Isolation in Space-Bounded Settings

We study the possibility of deterministic and randomness-efficient isolation in space-bounded models of computation: Can one efficiently reduce instances of computational problems to equivalent instances that have at most one solution? We present results for the NL-complete problem of reachability on digraphs, and for the LogCFL-complete problem of certifying acceptance on shallow semi-unbounded circuits. A common approach employs small weight assignments that make the solution of minimum weight unique. The Isolation Lemma and other known procedures use Omega(n) random bits to generate weights of individual bitlength O(log(n)). We develop a derandomized version for both settings that uses O(log(n)^{3/2}) random bits and produces weights of bitlength O(log(n)^{3/2}) in logarithmic space. The construction allows us to show that every language in NL can be accepted by a nondeterministic machine that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. Similarly, every language in LogCFL can be accepted by a nondeterministic machine equipped with a stack that does not count towards the space bound, that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. We also show that the existence of somewhat more restricted isolations for reachability on digraphs implies that NL can be decided in logspace with polynomial advice. A similar result holds for certifying acceptance on shallow semi-unbounded circuits and LogCFL

Revisiting the Equivalence Problem for Finite Multitape Automata

The decidability of determining equivalence of deterministic multitape automata (or transducers) was a longstanding open problem until it was resolved by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability yields a co_NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability, which follows the basic strategy of Harju and Karhumaki but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding language equivalence of deterministic multitape automata and, more generally, multiplicity equivalence of nondeterministic multitape automata. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ

Variety Membership Testing in Algebraic Complexity Theory

In this thesis, we study some of the central problems in algebraic complexity theory through the lens of the variety membership testing problem. In the first part, we investigate whether separations between algebraic complexity classes can be phrased as instances of the variety membership testing problem. For this, we compare some complexity classes with their closures. We show that monotone commutative single-(source, sink) ABPs are closed. Further, we prove that multi-(source, sink) ABPs are not closed in both the monotone commutative and the noncommutative settings. However, the corresponding complexity classes are closed in all these settings. Next, we observe a separation between the complexity class VQP and the closure of VNP. In the second part, we cover the blackbox polynomial identity testing (PIT) problem, and the rank computation problem of symbolic matrices, both phrasable as instances of the variety membership testing problem. For the blackbox PIT, we give a randomized polynomial time algorithm that uses the number of random bits that matches the information-theoretic lower bound, differing from it only in the lower order terms. For the rank computation problem, we give a deterministic polynomial time approximation scheme (PTAS) when the degrees of the entries of the matrices are bounded by a constant. Finally, we show NP-hardness of two problems on 3-tensors, both of which are instances of the variety membership testing problem. The first problem is the orbit closure containment problem for the action of GLk x GLm x GLn on 3-tensors, while the second problem is to decide whether the slice rank of a given 3-tensor is at most r

Revisiting the Equivalence Problem for Finite Multitape Automata

Abstract. The decidability of determining equivalence of deterministic multitape automata was a longstanding open problem until it was resolved by Harju and Karhumäki in the early 1990s. Their proof of decidability yields a co-NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability which follows the basic strategy of Harju and Karhumäki, but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding equivalence of deterministic multitape automata, as well as automata with transition weights in the field of rational numbers. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum