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

A Quantum Time-Space Lower Bound for the Counting Hierarchy

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real epsilon there exists a real c>1 such that either: MajMajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n^c, or MajSAT does not have a quantum algorithm with bounded two-sided error that runs in time n^d and space n^{1-\epsilon}. In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded two-sided error running in time n^{1+o(1)} and space n^{1-\epsilon} for any epsilon>0. The key technical novelty is a time- and space-efficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation.Comment: 25 page

Polynomial Identity Testing via Evaluation of Rational Functions

We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. In spite of the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas. We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition size of set-multi-linearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept we rederive known derandomization results based on the generator by Shpilka and Volkovich, and present a new application for read-once oblivious arithmetic branching programs that provably transcends the usual combinatorial techniques

Polynomial Identity Testing via Evaluation of Rational Functions

We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. Despite the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas. We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition class size of set-multilinearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept, we rederive known derandomization results based on the generator by Shpilka and Volkovich and present a new application for read-once oblivious algebraic branching programs.Comment: Appeared at ITCS 202

Query Complexity of Inversion Minimization on Trees

We consider the following computational problem: Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical psychology, with the evaluation of the Mann--Whitney statistic for detecting differences between distributions as a special case. We study the complexity of the problem in the comparison-query model, used for problems like sorting and selection. For many types of trees with $n$ leaves, we establish lower bounds close to the strongest known in the model, namely the lower bound of $\log_2(n!)$ for sorting $n$ items. We show: (a) $\log_2((\alpha(1-\alpha)n)!) - O(\log n)$ queries are needed whenever the tree has a subtree that contains a fraction $\alpha$ of the leaves. This implies a lower bound of $\log_2((\frac{k}{(k+1)^2}n)!) - O(\log n)$ for trees of degree $k$. (b) $\log_2(n!) - O(\log n)$ queries are needed in case the tree is binary. (c) $\log_2(n!) - O(k \log k)$ queries are needed for certain classes of trees of degree $k$, including perfect trees with even $k$. The lower bounds are obtained by developing two novel techniques for a generic problem $\Pi$ in the comparison-query model and applying them to inversion minimization on trees. Both techniques can be described in terms of the Cayley graph of the symmetric group with adjacent-rank transpositions as the generating set. Consider the subgraph consisting of the edges between vertices with the same value under $\Pi$. We show that the size of any decision tree for $\Pi$ must be at least: (i) the number of connected components of this subgraph, and (ii) the factorial of the average degree of the complementary subgraph, divided by $n$. Lower bounds on query complexity then follow by taking the base-2 logarithm.Comment: 54 pages, 18 figures, full version of paper appearing in the Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithm

08381 Abstracts Collection -- Computational Complexity of Discrete Problems

From the 14th of September to the 19th of September, the Dagstuhl Seminar 08381 ``Computational Complexity of Discrete Problems\u27\u27 was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work as well as open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this report. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available