Theory of Computing Systems最新文献

The notion of a quasi-order generated by a homomorphism from the semigroup of all words onto a finite ordered semigroup was introduced by Bucher et al. (Theor. Comput. Sci. 40, 131–148 1985). It naturally occurred in their studies of derivation relations associated with a given set of context-free rules, and they asked a crucial question, whether the resulting relation is a well quasi-order. We answer this question in the case of the quasi-order generated by a semigroup homomorphism. We show that the answer does not depend on the homomorphism, but it is a property of its image. Moreover, we give an algebraic characterization of those finite semigroups for which we get well quasi-orders. This characterization completes the structural characterization given by Kunc (Theor. Comput. Sci. 348, 277–293 2005) in the case of semigroups ordered by equality. Compared with Kunc’s characterization, the new one has no structural meaning, and we explain why that is so. In addition, we prove that the new condition is testable in polynomial time.

In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size (k = sqrt{n}) planted in a random (G(n, frac{1}{2})) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size (k=frac{n}{2}) in a G(n, p) graph with (p = n^{-frac{1}{2}}), there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms.

We consider a setting with agents that have preferences over alternatives and are partitioned into disjoint districts. The goal is to choose one alternative as the winner using a mechanism which first decides a representative alternative for each district based on a local election with the agents therein as participants, and then chooses one of the district representatives as the winner. Previous work showed bounds on the distortion of a specific class of deterministic plurality-based mechanisms depending on the available information about the preferences of the agents in the districts. In this paper, we first consider the whole class of deterministic mechanisms and show asymptotically tight bounds on their distortion. We then initiate the study of the distortion of randomized mechanisms in distributed voting and show bounds based on several informational assumptions, which in many cases turn out to be tight. Finally, we also experimentally compare the distortion of many different mechanisms of interest using synthetic and real-world data.

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.