Theory of Computing Systems最新文献

In 2015, Haviv introduced the Remote set problem (RSP) and studied the complexity of the covering radius problem (CRP), which is a classical problem in lattices. The RSP aims to identify a set containing a point that is sufficiently distant from a given lattice (pmb {mathcal {L}}). It introduced a new method for analyzing the complexity of CRP. An open question in RSP is whether we can obtain the approximation factor (gamma =1/2). This paper investigates this question and proposes a probabilistic polynomial-time algorithm for RSP with an approximation factor of (1/2-1/(clambda ^{(p)}_n)), where (cin mathbb {Z}^{+}) and (lambda ^{(p)}_n) is the n-th successive minima in lattice under (l_p)-norm. For a given lattice (pmb {mathcal {L}}) with rank n and positive integer d, our algorithm outputs a set S of size d in polynomial time. This set S includes a point at least ((frac{1}{2}-frac{1}{clambda ^{(p)}_n}){{rho }^{(p)}}(pmb {mathcal {L}})) from lattice (pmb {mathcal {L}}) with a probability greater than (1-1/2^d). Here, c is a positive integer and (rho ^{(p)}(pmb {mathcal {L}})) denotes the covering radius of (pmb {mathcal {L}}) in (l_p)-norm((1le ple infty )). Based on this, we obtain that (text {GAPCRP}_{2+1/2^{O(n)}}) belongs to the complexity class coRP, and we provide new reductions from GAPCRP to GAPCVP.

The paper studies (varvec{Sigma ^0_n})-computable families ((varvec{ngeqslant 2})) and their numberings. It is proved that any non-trivial (varvec{Sigma ^0_n})-computable family has a complete with respect to any of its elements (varvec{Sigma ^0_n})-computable non-principal numbering. It is established that if a (varvec{Sigma ^0_n})-computable family is not principal, then any of its (varvec{Sigma ^0_n})-computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal (varvec{Sigma ^0_n})-computable numberings. It is also shown that for any (varvec{Sigma ^0_n})-computable numbering (varvec{nu }) of a (varvec{Sigma ^0_n})-computable non-principal family there exists its (varvec{Sigma ^0_n})-computable numbering that is incomparable with (varvec{nu }). If a non-trivial (varvec{Sigma ^0_n})-computable family contains the least and greatest elements under inclusion, then for any of its (varvec{Sigma ^0_n})-computable non-principal non-least numberings (varvec{nu }) there exists a (varvec{Sigma ^0_n})-computable numbering of the family incomparable with (varvec{nu }). In particular, this is true for the family of all (varvec{Sigma ^0_n})-sets and for the families consisting of two inclusion-comparable (varvec{Sigma ^0_n})-sets (semilattices of the (varvec{Sigma ^0_n})-computable numberings of such families are isomorphic to the semilattice of (varvec{m})-degrees of (varvec{Sigma ^0_n})-sets).

The well-known Cluster Vertex Deletion problem (cluster-vd) asks for a given graph G and an integer k whether it is possible to delete a set S of at most k vertices of G such that the resulting graph (G-S) is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs H for which cluster-vd on H-free graphs is polynomially solvable and for which it is (textsf{NP})-complete. Moreover, in the (textsf{NP})-completeness cases, cluster-vd cannot be solved in sub-exponential time in the vertex number of the H-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of cluster-vd, the Connected Cluster Vertex Deletion problem (connected cluster-vd), in which the set S has to induce a connected subgraph of G. It turns out that connected cluster-vd admits the same complexity dichotomy for H-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on H-free graphs.

Abstract

We introduce the infix inclusion problem of two languages S and T that decides whether or not S is a subset of the set of all infixes of T. This problem is motivated by the need for identifying malicious computation patterns according to their semantics, which are often disguised with additional sub-patterns surrounding information. In other words, malicious patterns are embedded as an infix of the whole pattern. We examine the infix inclusion problem for the case where a source S and a target T are finite, regular or context-free languages. We prove that the problem is 1) co-NP-complete when one of the languages is finite, 2) PSPACE-complete when both S and T are regular, 3) EXPTIME-complete when S is context-free and T is regular, 4) undecidable when S is either regular or context-free and T is context-free and 5) undecidable when one of S and T is in a language class where the emptiness of its languages is undecidable, even if the other is finite. We, furthermore, explore the infix inclusion problem for visibly pushdown languages, a subclass of context-free languages.

In this paper, we analyze the copy complexity of unsatisfiable Horn constraint systems, under the ADD refutation system. Recall that a linear constraint of the form (sum _{i=1}^{n} a_{i}cdot x_{i} ge b), is said to be a horn constraint if all the (a_{i} in {0,1,-1}) and at most one of the (a_{i})s is positive. A conjunction of such constraints is called a Horn constraint system (HCS). Horn constraints arise in a number of domains including, but not limited to, program verification, power systems, econometrics, and operations research. The ADD refutation system is both sound and complete. Additionally, it is the simplest and most natural refutation system for refuting the feasibility of a system of linear constraints. The copy complexity of an infeasible linear constraint system (not necessarily Horn) in a refutation system, is the minimum number of times each constraint needs to be replicated, in order to obtain a read-once refutation. We show that for an HCS with n variables and m constraints, the copy complexity is at most (2^{n-1}), in the ADD refutation system. Additionally, we analyze bounded-width HCSs from the perspective of copy complexity. Finally, we provide an empirical analysis of an integer programming formulation of the copy complexity problem in HCSs. (An extended abstract was published in FroCos 2021 [26].)

We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is b _n = n!. Our main result is that the sequence u _n = n ⁿ is not polynomial recursive.

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.