Theory of Computing Systems最新文献

In MAXSPACE, given a set of ads (mathcal {A}), one wants to schedule a subset ({mathcal {A}'subseteq mathcal {A}}) into K slots ({B_1, dots , B_K}) of size L. Each ad ({A_i in mathcal {A}}) has a size (s_i) and a frequency (w_i). A schedule is feasible if the total size of ads in any slot is at most L, and each ad ({A_i in mathcal {A}'}) appears in exactly (w_i) slots and at most once per slot. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We consider a generalization called MAXSPACE-R for which an ad (A_i) also has a release date (r_i) and may only appear in a slot (B_j) if ({j ge r_i}). For this variant, we give a 1/9-approximation algorithm. Furthermore, we consider MAXSPACE-RDV for which an ad (A_i) also has a deadline (d_i) (and may only appear in a slot (B_j) with (r_i le j le d_i)), and a value (v_i) that is the gain of each assigned copy of (A_i) (which can be unrelated to (s_i)). We present a polynomial-time approximation scheme for this problem when K is bounded by a constant. This is the best factor one can expect since MAXSPACE is strongly NP-hard, even if (K = 2).

Studies of issues related to computability and computational complexity involve the use of a model of computation. Central in such a model are computational processes. Processes of this kind can be described using an imperative process algebra based on ACP (Algebra of Communicating Processes). In this paper, it is investigated whether the imperative process algebra concerned can play a role in the field of models of computation. It is demonstrated that the process algebra is suitable to describe in a mathematically precise way models of computation corresponding to existing models based on sequential, asynchronous parallel, and synchronous parallel random access machines as well as time and work complexity measures for those models.

The descriptional complexity of basic operations on regular languages using 1-limited automata, a restricted version of one-tape Turing machines, is investigated. When simulating operations on deterministic finite automata with deterministic 1-limited automata, the sizes of the resulting devices are polynomial in the sizes of the simulated machines. The situation is different when the operations are applied to deterministic 1-limited automata: while for boolean operations the simulations remain polynomial, for product, star, and reversal they cost exponential in size. The costs for product and star do not reduce if the given machines are sweeping two-way deterministic finite automata. These bounds are tight.

Abstract

The homogeneous weight (metric) is useful in the construction of codes over a ring of integers (mathbb {Z}_{p^l}) (p prime and (l ge 1) an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be (mathbb {Z}_{4}) . This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of n-tuples over (mathbb {Z}_{p^l}) . Necessary and sufficient conditions for existence of an (n, k) linear code over (mathbb {Z}_{p^l}) correcting the error patterns with respect to the homogeneous weight are derived.

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.