Theory of Computing Systems最新文献

A pumping lemma for a class of languages (varvec{mathcal {C}}) is often used to show particular languages are not in (varvec{mathcal {C}}). In contrast, we show that a pumping lemma for a class of languages (varvec{mathcal {C}}) can be used to study the computational complexity of the predicate “(in varvec{mathcal {C}})” via highly efficient many-one reductions. In this paper, we use extended regular expressions (EXREGs, introduced in Câmpeanu et al. (Int. J. Foundations Comput. Sci. 14(6), 1007–1018, 2003)) as an example to illustrate the proof technique and establish the complexity of the predicate “is an EXREG language” for several classes of languages. Due to the efficiency of the reductions, both productiveness (a stronger form of non-recursive enumerability) and complexity results can be obtained simultaneously. For example, we show that the predicate “is an EXREG language” is productive (hence, not recursively enumerable) for context-free grammars, and is Co-NEXPTIME-hard for context-free grammars generating bounded languages. The proof technique is easy to use and requires only a few conditions. This suggests that for any class of languages (varvec{mathcal {C}}) having a pumping lemma, the language class comparison problems (e.g., does a given context-free grammar generate a language in (varvec{mathcal {C}})?) are almost guaranteed to be hard. So, pumping lemmas sometimes could be “harmful” when studying computational complexity results.

We study the problem of placing wildlife crossings, such as green bridges, over human-made obstacles to challenge habitat fragmentation. The main task herein is, given a graph describing habitats or routes of wildlife animals and possibilities of building green bridges, to find a low-cost placement of green bridges that connects the habitats. We develop three problem models for this task and study them from a computational complexity and parameterized algorithmics perspective.

Abstract

Many Boolean functions that need to be encoded as CNF in practice, have only exponential size CNF representations. To avoid this effect, one usually introduces nondeterministic variables. For example, whereas the minimum number of clauses in a CNF computing the parity function (x_1oplus x_2 oplus cdots oplus x_n) is  (2^{n-1}) , one can use (n-1) nondeterministic variables to get a CNF encoding with 4n clauses. In this paper, we prove tradeoffs between various parameters (the number of clauses, the width of clauses, and the number of nondeterministic variables) of CNF encodings of various symmetric functions. In particular, we show that a folklore way of encoding parity as CNF is provably optimal. We do this by using a tight connection between CNF encodings and depth-3 circuits. This connection shows that CNF encodings is an interesting computational model for Boolean functions: on the one hand, it is routinely used in practice when translating a computational problem to a format acceptable by a SAT solver, on the other hand, lower bounds on the size of CNF encodings imply depth-3 circuit lower bounds.

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.