Theory of Computing Systems最新文献

The rational index (rho _L) of a language L is an integer function, where (rho _L(n)) is the maximum length of the shortest string in (L cap R), over all regular languages R recognized by n-state nondeterministic finite automata (NFA). This paper investigates the rational index of languages defined by grammars with bounded parse tree dimension: this is a numerical measure of the amount of branching in a tree (with trees in a linear grammar having dimension 1). For context-free grammars, a grammar with tree dimension bounded by d has rational index at most (O(n^{2d})), and it is known from the literature that there exists a grammar with rational index (Theta (n^{2d})). In this paper, it is shown that for multi-component grammars with at most k components (k-MCFG) and with a tree dimension bounded by d, the rational index is at most (O(n^{2kd})), where the constant depends on the grammar, and there exists such a grammar with rational index (frac{k}{2^{kd^2 - kd -2k -1} cdot (8k+1)^{2kd}} n^{2kd}). Also, for the case of ordinary context-free grammars, a more precise lower bound (frac{1}{2^{d^2 + d - 3} 3^{2d}} n^{2d}) is established.

We study the problem of computing an approximate maximum cardinality matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, the edges of the input graph (G = (V,E)) are given as a stream (e_1, ldots , e_m), and the algorithm is allowed to make a single pass over this stream while using (O(ntext {polylog}(n))) space ((m = |E|) and (n = |V|)). If the order of edges is adversarial, a simple single-pass greedy algorithm yields a 1/2-approximation in O(n) space; achieving a better approximation in adversarial streams remains an elusive open question. A line of recent work shows that one can improve upon the 1/2-approximation if the edges of the stream arrive in a random order. The state of the art for this model is two-fold: Assadi et al. [SODA 2019] show how to compute a (frac{2}{3}) ((sim .66))-approximate matching, but the space requirement is (O(n^{1.5}text {polylog}(n))). Very recently, Farhadi et al. [SODA 2020] presented an algorithm with the desired space usage of (O(ntext {polylog}(n))), but a worse approximation ratio of (frac{6}{11}) ((sim .545)), or (frac{3}{5}) ((=.6)) in bipartite graphs. In this paper, we present an algorithm that computes a (frac{2}{3}(sim .66))-approximate matching using only (O(nlog (n))) space, improving upon both results above. We also note that for adversarial streams, a lower bound of Kapralov [SODA 2013] shows that any algorithm that achieves a (1-frac{1}{e})((sim .63))-approximation requires ((n^{1+Omega (1/log log (n))})) space; recent follow-up work by the same author improved this lower bound to (1+ln (2) sim .59) [SODA 2021]. As a consequence, both our result and the earlier result of Farhadi et al. prove that the problem of computing a maximum matching is strictly easier in random-order streams than in adversarial ones.

We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow.

We show that any Boolean function with approximate rank r can be computed by bounded-error quantum protocols without prior entanglement of complexity (O( sqrt{r} log r)). In addition, we show that any Boolean function with approximate rank r and discrepancy (delta ) can be computed by deterministic protocols of complexity O(r), and private coin bounded-error randomized protocols of complexity (O((frac{1}{delta })^2 + log r)). Our deterministic upper bound in terms of approximate rank is tight up to constant factors, and the dependence on discrepancy in our randomized upper bound is tight up to taking square-roots. Our results can be used to obtain lower bounds on approximate rank. We also obtain a strengthening of Newman’s theorem with respect to approximate rank.

Word equations are equations (alpha doteq beta ) where (alpha ) and (beta ) are words consisting of letters from some alphabet (Sigma ) and variables from a set X. Recently, there has been substantial interest in the context of string solving in logics combining word equations with other kinds of constraints on words such as (regular) language membership (regular constraints) and arithmetic over string lengths (length constraints). We consider the expressive power of such logics by looking at the set of all values a single variable might take as part of a satisfying assignment for a given formula. Hence, each formula-variable pair defines a formal language, and each logic defines a class of formal languages. We consider logics arising from combining word equations with either length constraints, regular constraints, or both. We also consider word equations with visibly pushdown language membership constraints as a generalisation of the combination of regular and length constraints. We show that word equations with visibly pushdown membership constraints are sufficient to express all recursively enumerable languages and hence satisfiability is undecidable in this case. We then establish a strict hierarchy involving the other combinations. We also provide a complete characterisation of when a thin regular language is expressible by word equations (alone) and some further partial results for regular languages in the general case.

On sparse graphs, Roditty and Williams [2013] proved that no (varvec{O(n^{2-varepsilon })})-time algorithm achieves an approximation factor smaller than (varvec{frac{3}{2}}) for the diameter problem unless SETH fails. In this article, we solve an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for all eccentricities in median graphs with bounded dimension (varvec{d}), i.e. the dimension of the largest induced hypercube. This prerequisite on (varvec{d}) is not necessary anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is (varvec{O(n^{1.6456}log ^{O(1)} n)}). We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerates all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time (varvec{O(2^{3d}nlog ^{O(1)}n)}).

In this paper, we examine variants of the partial vertex cover problem from the perspective of parameterized algorithms. Recall that in the classical vertex cover problem (VC), we are given a graph (mathbf{G = langle V, E rangle }) and a number k and asked if we can cover all of the edges in (textbf{E}), using at most k vertices from (textbf{V}). The partial vertex cover problem (PVC) is a more general version of the VC problem in which we are given an additional parameter (k'). We then ask the question of whether at least (k') of the edges in (textbf{E}) can be covered using at most k vertices from (textbf{V}). Note that the VC problem is a special case of the PVC problem when (k'=|textbf{E}|). In this paper, we study the weighted generalizations of the PVC problem. This is called the weighted partial vertex cover problem (WPVC). In the WPVC problem, we are given two parameters R and L, associated respectively with the vertex set (textbf{V}) and edge set (textbf{E}) of the graph (textbf{G}) respectively. Additionally, we are given non-negative integral weight functions for the vertices and the edges. The goal then is to cover edges of total weight at least L, using vertices of total weight at most R. This paper studies several variants of the PVC and WPVC problems and establishes new results from the perspective of fixed-parameter tractability and W[1]-hardness. We also introduce a new problem called the partial vertex cover with matching constraints and show that it is Fixed-Parameter Tractable (FPT) for a certain class of graphs. Finally, we show that the WPVC problem is APX-complete for bipartite graphs.