Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial). In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds

Counting Euler Tours in Undirected Bounded Treewidth Graphs

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize $\#SAC^1$) is relatively easy, establishing a uniform $\#SAC^1$ bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound claimed in the previous version which has been subsequently remove

Complexity of Restricted Variants of Skolem and Related Problems

Given a linear recurrence sequence (LRS), the Skolem problem, asks whether it ever becomes zero. The decidability of this problem has been open for several decades. Currently decidability is known only for LRS of order upto 4. For arbitrary orders (i.e., number of terms the n-th depends on), the only known complexity result is NP-hardness by a result of Blondel and Portier from 2002. In this paper, we give a different proof of this hardness result, which is arguably simpler and pinpoints the source of hardness. To demonstrate this, we identify a subclass of LRS for which the Skolem problem is in fact NP-complete. We show the generic nature of our lower-bound technique by adapting it to show stronger lower bounds of a related problem that encompasses many known decision problems on linear recurrent sequences

On the Complexity of Value Iteration

Value iteration is a fundamental algorithm for solving Markov Decision Processes (MDPs). It computes the maximal n-step payoff by iterating n times a recurrence equation which is naturally associated to the MDP. At the same time, value iteration provides a policy for the MDP that is optimal on a given finite horizon n. In this paper, we settle the computational complexity of value iteration. We show that, given a horizon n in binary and an MDP, computing an optimal policy is EXPTIME-complete, thus resolving an open problem that goes back to the seminal 1987 paper on the complexity of MDPs by Papadimitriou and Tsitsiklis. To obtain this main result, we develop several stepping stones that yield results of an independent interest. For instance, we show that it is EXPTIME-complete to compute the n-fold iteration (with n in binary) of a function given by a straight-line program over the integers with max and + as operators. We also provide new complexity results for the bounded halting problem in linear-update counter machines

Graph Properties in Node-Query Setting: Effect of Breaking Symmetry

The query complexity of graph properties is well-studied when queries are on the edges. We investigate the same when queries are on the nodes. In this setting a graph G = (V,E) on n vertices and a property P are given. A black-box access to an unknown subset S of V is provided via queries of the form "Does i belong to S?". We are interested in the minimum number of queries needed in the worst case in order to determine whether G[S] - the subgraph of G induced on S - satisfies P. Our primary motivation to study this model comes from the fact that it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on the hereditary graph properties - properties that are closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n. We show that in the absence of any symmetry on G it can fall as low as O(n^{1/(d + 1)}) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: Can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Omega(n^{1/k}) for a constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Omega(log(n)*log(log(n))) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdos and Rado

Cyclotomic Identity Testing and Applications

We consider the cyclotomic identity testing problem: given a polynomial $f(x_1,\ldots,x_k)$, decide whether $f(\zeta_n^{e_1},\ldots,\zeta_n^{e_k})$ is zero, for $\zeta_n = e^{2\pi i/n}$ a primitive complex $n$-th root of unity and integers $e_1,\ldots,e_k$. We assume that $n$ and $e_1,\ldots,e_k$ are represented in binary and consider several versions of the problem, according to the representation of $f$. For the case that $f$ is given by an algebraic circuit we give a randomized polynomial-time algorithm with two-sided errors, showing that the problem lies in BPP. In case $f$ is given by a circuit of polynomially bounded syntactic degree, we give a randomized algorithm with two-sided errors that runs in poly-logarithmic parallel time, showing that the problem lies in BPNC. In case $f$ is given by a depth-2 $\Sigma\Pi$ circuit (or, equivalently, as a list of monomials), we show that the cyclotomic identity testing problem lies in NC. Under the generalised Riemann hypothesis, we are able to extend this approach to obtain a polynomial-time algorithm also for a very simple subclass of depth-3 $\Sigma\Pi\Sigma$ circuits. We complement this last result by showing that for a more general class of depth-3 $\Sigma\Pi\Sigma$ circuits, a polynomial-time algorithm for the cyclotomic identity testing problem would yield a sub-exponential-time algorithm for polynomial identity testing. Finally, we use cyclotomic identity testing to give a new proof that equality of compressed strings, i.e., strings presented using context-free grammars, can be decided in coRNC: randomized NC with one-sided errors

Multiplicity Problems on Algebraic Series and Context-Free Grammars

In this paper we obtain complexity bounds for computational problems on algebraic power series over several commuting variables. The power series are specified by systems of polynomial equations: a formalism closely related to weighted context-free grammars. We focus on three problems -- decide whether a given algebraic series is identically zero, determine whether all but finitely many coefficients are zero, and compute the coefficient of a specific monomial. We relate these questions to well-known computational problems on arithmetic circuits and thereby show that all three problems lie in the counting hierarchy. Our main result improves the best known complexity bound on deciding zeroness of an algebraic series. This problem is known to lie in PSPACE by reduction to the decision problem for the existential fragment of the theory of real closed fields. Here we show that the problem lies in the counting hierarchy by reduction to the problem of computing the degree of a polynomial given by an arithmetic circuit. As a corollary we obtain new complexity bounds on multiplicity equivalence of context-free grammars restricted to a bounded language, language inclusion of a nondeterministic finite automaton in an unambiguous context-free grammar, and language inclusion of a non-deterministic context-free grammar in an unambiguous finite automaton.Comment: full technical report of a LICS'23 pape