We investigate when non-dictatorial aggregation is possible from an algorithmic perspective, where non-dictatorial aggregation means that the votes cast by the members of a society can be aggregated in such a way that there is no single member of the society that always dictates the collective outcome. We consider the setting in which the members of a society take a position on a fixed collection of issues, where for each issue several different alternatives are possible, but the combination of choices must belong to a given set X of allowable voting patterns. Such a set X is called a possibility domain if there is an aggregator that is non-dictatorial, operates separately on each issue, and returns values among those cast by the society on each issue. We design a polynomial-time algorithm that decides, given a set X of voting patterns, whether or not X is a possibility domain. Furthermore, if X is a possibility domain, then the algorithm constructs in polynomial time a non-dictatorial aggregator for X. Furthermore, we show that the question of whether a Boolean domain X is a possibility domain is in NLOGSPACE. We also design a polynomial-time algorithm that decides whether X is a uniform possibility domain, that is, whether X admits an aggregator that is non-dictatorial even when restricted to any two positions for each issue. As in the case of possibility domains, the algorithm also constructs in polynomial time a uniform non-dictatorial aggregator, if one exists. Then, we turn our attention to the case where X is given implicitly, either as the set of assignments satisfying a propositional formula, or as a set of consistent evaluations of a sequence of propositional formulas. In both cases, we provide bounds to the complexity of deciding if X is a (uniform) possibility domain. Finally, we extend our results to four types of aggregators that have appeared in the literature: generalized dictatorships, whose outcome is always an element of their input, anonymous aggregators, whose outcome is not affected by permutations of their input, monotone, whose outcome does not change if more individuals agree with it and systematic, which aggregate every issue in the same way.

In the field of Judgment Aggrgation, a domain, that is a subset of a Cartesian power of $\{0,1\}$, is considered to reflect abstract rationality restrictions on vectors of two-valued judgments on a number of issues. We are interested in the ways we can aggregate the positions of a set of individuals, whose positions over each issue form vectors of the domain, by means of unanimous (idempotent) functions, whose output is again an element of the domain. Such functions are called non-dictatorial, when their output is not simply the positions of a single individual. Here, we consider domains admitting various kinds of non-dictatorial aggregators, which reflect various properties of majority aggregation: (locally) non-dictatorial, generalized dictatorships, anonymous, monotone, StrongDem and systematic. We show that interesting and, in some sense, democratic voting schemes are always provided by domains that can be described by propositional formulas of specific syntactic types we define. Furthermore, we show that we can efficiently recognize such formulas and that, given a domain, we can both efficiently check if it is described by such a formula and, in case it is, construct it. Our results fall in the realm of classical results concerning the syntactic characterization of domains with specific closure properties, like domains closed under logical AND which are the models of Horn formulas. The techniques we use to obtain our results draw from judgment aggregation as well as propositional logic and universal algebra.

We consider the problem of aggregating votes cast by a society on a fixed set of issues, where each member of the society may vote for one of several positions on each issue, but the combination of votes on the various issues is restricted to a set of feasible voting patterns. We follow the aggregation framework used by Dokow and Holzman [Aggregation of non-binary evaluations, Advances in Applied Mathematics , 45:4, 487--504, 2010], in which both preference aggregation and judgment aggregation can be cast. We require the aggregation to be independent on each issue, and also supportive, i.e., for every issue, the corresponding component of every aggregator, when applied to a tuple of votes, must take as value one of the votes in that tuple. We prove that, in such a setup, non-dictatorial aggregation of votes in a society of an arbitrary size is possible if and only if either there is a non-dictatorial aggregator for two voters or there is an aggregator for three voters such that, for each issue, the corresponding component of the aggregator, when restricted to two-element sets of votes, is a majority operation or a minority operation. We then introduce a notion of a uniform non-dictatorial aggregator, which is an aggregator such that on every issue, and when restricted to arbitrary two-element subsets of the votes for that issue, it differs from all projection functions. We first give a characterization of sets of feasible voting patterns that admit a uniform non-dictatorial aggregator. After this and by making use of Bulatov’s dichotomy theorem for conservative constraint satisfaction problems, we connect social choice theory with the computational complexity of constraint satisfaction by proving that if a set of feasible voting patterns has a uniform non-dictatorial aggregator of some arity, then the multi-sorted conservative constraint satisfaction problem on that set (with each issue representing a different sort) is solvable in polynomial time; otherwise, it is NP-complete.

We show that a randomly chosen $3$-CNF formula over $n$ variables with clauses-to-variables ratio at least $4.4898$ is asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was $4.506$. The first such bound, independently discovered by many groups of researchers since 1983, was $5.19$. Several decreasing values between $5.19$ and $4.506$ were published in the years between. The probabilistic techniques we use for the proof are, we believe, of independent interest.

Abstract We consider the problem of searching for a piece of information in a fully interconnected computer network (also called a complete network orclique) by exploiting advice about its location from the network nodes. Each node contains a database that “knows” what kind of documents or information are stored in other nodes (e.g., a node could be a Web server that answers queries about documents stored on the Web). The databases in each node, when queried, provide a pointer that leads to the node that contains the information. However, this information is up‐to‐date (or correct) with some bounded probability. While, in principle, one may always locate the information by simply visiting the network nodes in some prescribed ordering, this requires a time complexity in the order of the number of nodes of the network. In this paper, we provide algorithms for locating an information node in the complete communication network, which take advantage ofadvicegiven from network nodes. The nodes may either give correct advice, by pointing directly to the information node, or give wrong advice, by pointing elsewhere. On the lower‐bounds' side, we show that no fixed‐memory (i.e., with memory independent of the network size) deterministic algorithm may locate the information node in a constant (independent of the network size) expected number of steps. Moreover, ifp= ω(1/n) is the probability that a node of ann‐node clique gives correct advice, we show that no algorithm may locate the information node in an expected number of steps less than 1/p−o(1). To study how the expected number of steps is affected by the amount of memory allowed to the algorithms, we give a memoryless randomized algorithm with expected number of steps 4/p+o(1/p) +o(1) and a 1‐bit randomized algorithm requiring on the average at most 2/p+o(1) steps. In addition, in the memoryless case, we also prove a 4/plower bound for the expected number of steps in the case where the nodes giving faulty advice may decide on the content of this advice in any possible way and not merely at random (adversarialfault model). Finally, for the case where faulty nodes behave randomly, we give an optimal, unlimited memory deterministic algorithm with expected number of steps bounded from above by 1/p+o(1/p) + 1. © 2003 Wiley Periodicals, Inc.

The inference problem for propositional circumscription is known to be highly intractable and, in fact, harder than the inference problem for classical propositional logic. More precisely, in its full generality this problem in /spl Pi//sub 2//sup P/-complete, which means that it has the same inherent computational complexity as the satisfiability problem for quantified Boolean formulas with two alternations (universal-existential) of quantifiers. We use T.J. Schaefer's (1978) framework of generalized satisfiability problems to study the family of all restricted cases of the inference problem for propositional circumscription. Our main result fields a complete classification of the "truly hard"(/spl Pi//sub 2//sup P/-complete) and the "easier" cases of this problem (reducible to the inference problem for classical propositional logic). Specifically, we establish a dichotomy theorem which asserts that each such restricted case is either /spl Pi//sub 2//sup P/-complete or is in co-NP. Moreover, we provide efficiently checkable criteria that tell apart the "truly hard" cases from the "easier" ones.

In this paper we examine the problem of searching for some information item in the nodes of a fully interconnected computer network, where each node contains information relevant to some topic as well as links to other network nodes that also contain information, not necessarily related to locally kept information. These links are used to facilitate the Internet users and mobile software agents that try to locate specific pieces of information. However, the links do not necessarily point to nodes containing information of interest to the user or relevant to the aims of the mobile agent. Thus an element of uncertainty is introduced. For example, when an Internet user or some search agent lands on a particular network node, they see a set of links that point to information that is, supposedly, relevant to the current search. Therefore, we can assume that a link points to relevant information with some unknown probability $p$ that, in general, is related to the number of nodes in the network (intuitively, as the network grows, this probability tends to zero since adding more nodes to the network renders some extant links less accurate or obsolete). Consequently, since there is uncertainty as to whether the links contained in a node's Web page are correct or not, a search algorithm cannot rely on following the links systematically since it may end up spending too much time visiting nodes that contain irrelevant information. In this work, we will describe and analyze a search algorithm that is only allowed to transfer a fixed amount of memory along communication links as it visits the network nodes. The algorithm is, however, allowed to use one bit of memory at each node as an ‘already visited’ flag. In this way the algorithm has its memory distributed to the network nodes, avoiding overloading the network links as it moves from node to node searching for the information. We work on fully interconnected networks for simplicity reasons and, moreover, because according to some recent experimental evidence, such networks can be considered to be a good approximation of the current structure of the World Wide Web.

Let X1,…,Xc be variables which together constitute a composite register. These variables are shared by a number of processes which operate in a totally asynchronous and wait-free manner. An operation by a process on the composite register is either a write to one of the variables or a read of the values of all variables. All operations are required to be atomic, i.e. an execution of any number of them (including reads) must be linearizable, in a way consistent with the values returned by the reads. In a single reader composite register no two reads can concurrently access the composite register. We give a new protocol implementing a single reader composite register for the case when there is a single writer per variable. Our construction uses time-stamps that may take values as large as the number of operations performed. The advantages of our construction over previous (bounded time-stamps) solutions are: (i) Both the protocol and its formal correctness proof are easy to understand. (ii) The time complexity of an operation of our construction (i.e. the number of its sub-operations) and the number of the subregisters used in our construction are at most equal to the number of processes that can concurrently access the composite register.

The linkage of a graph is defined to be the maximum mm-degree of any of its subgraphs It is known that the linkage of a graph is equal to its width: for an arbitrary linear ordering of the vertices of the graph, consider the maximum, with respect to any vertex v, of the number of vertices connected with v and preceding it in the ordering; the width of the graph is the minimum of these maxima over all possible linear orderings. Width has been used in artificial intelligence in the context of constraint satisfaction problems (CSPs). A more general notion is defined by considering not the number of vertices preceding and connected with v but rather the least number of vertices preceding and connected with any cluster of at most j consecutive vertices extending to the right up to v (j is a given integer). The graph parameter thus defined is called j-width. No efficient algorithm was known for computing the j-width. In this paper, we introduce a graph parameter depending on j that refers to the subgraphs of the graph and generalizes the notion of linkage. We prove the min–max theorem that this graph parameter, which we call j-linkage, is equal to j-width, and we then give a polynomial-time algorithm for computing it (for constant j). We also find tight lower and upper bounds for the j-linkage (equivalently, the j-width) of graphs with given numbers of vertices and edges. It is interesting to note that a lower bound for the width of a graph had been found by Erdös; as we show, however, that bound is not tight. Moreover, we prove that our lower bound for width is also a tight lower bound for treewidth, pathwidth, and bandwidth, graph parameters that may be arbitrarily larger than width. Finally, we show that computing the j-linkage is a P-complete problem, whereas we prove that approximating it is a threshold problem: it is in NC for approximation factors $ < {1 / {(2j)}}$, and it is P-complete for approximation factors $ > {1 / 2}$.

We address the problem of reading several variables (components) X/sub 1/,...,X/sub c/, all in one atomic operation, by only one process, called the reader, while each of these variables are being written by a set of writers. All operations (i.e., both reads and writes) are assumed to be totally asynchronous and wait-free. For this problem, only algorithms that require at best quadratic time and space complexity can be derived from the existing literature. (The time complexity of a construction is the number of suboperations of a high-level operation and its space complexity is the number of atomic shared variables it needs) In this paper, we provide a deterministic protocol that has linear (in the number of processes) space complexity, linear time complexity for a read operation, and constant time complexity for a write. Our solution does not make use of time-stamps. Rather, it is the memory location where a write writes that differentiates it from the other writes. Also, introducing randomness in the location where the reader gets the value that it returns, we get a conceptually very simple probabilistic algorithm. This algorithm has an overwhelmingly small, controllable probability of error. Its space complexity, and also the time complexity of a read operation, are sublinear. The time complexity of a write is constant. On the other hand, under the Archimedean time assumption, we get a protocol whose time and space complexity do not depend on the number of writers, but are linear in the number of components only. (The time complexity of a write operation is still constant.).>

This paper shows that the problem of testing whether a graph G contains an induced subgraph of vertex (edge) connectivity at least k is P-complete for any fixed $k \geqslant 3$. Moreover, if $k_{\max } $ is the largest vertex (edge) connectivity of any subgraph of G, it is shown that unless ${\text{P}} = {\text{NC}}$ there is no NC algorithm that approximates $k_{\max } $ within any approximation factor $\frac{1}{2} < c < 1$ (such an algorithm is by definition one that outputs a number in the interval $[ck_{\max } ,k_{\max } ]$). In contrast, it is known that the problem of finding the Tutte (triconnected) components of G (i.e., the maximal subgraphs of G such that for any four vertices in any of them, any two of these vertices can be connected by a path in G that avoids the other two) is in NC. On the positive side, it is shown, by proving extremal graph results, that the maximum k for which there is a k-edge-connected induced subgraph of G can be approximated in NC for any approximation factor strictly less than $\frac{1}{2}$ and that the same is true for vertex connectivity for any approximation factor strictly less than $\frac{1}{4}$.

A well-known method for interpreting planar projections (images) of three-dimensional polyhedra is to label their lines by the Clowes-Huffman scheme. However, the question of whether there is such a labeling has been shown to be NP-complete. A linear-in-time algorithm is given that answers the labelability question under the assumption that some information is known about those edges of the polyhedron both of whose faces are visible. In many cases, this information can be derived from the image itself. Moreover, the algorithm has an effective parallel version, i.e. with polynomially many processors it can be executed in time polynomial in log n.>

It is shown that the problem of testing whether a graph G contains a vertex- (edge-) connected induced subgraph of cardinality k is P-complete for any fixed k>or=3. Moreover, it is shown that approximating within a factor c>1/2 the maximum d for which there is a d-vertex-(d-edge-) connected induced subgraph of G is not in NC, unless P=NC. In contrast, it is known that the problem of finding the Tutte (triconnected) components of G is in NC. On the positive side, it is shown by proving extremal-graph results, that the maximum d for which there is a d-edge-connected induced subgraph of G can be approximated in NC within any factor c>

Article Free Access Share on Lower bounds and efficient algorithms for multiprocessor scheduling of dags with communication delays Authors: H. Jung Mathematics Dept., Humboldt Univ., GDR Mathematics Dept., Humboldt Univ., GDRView Profile , L. Kirousis Computer Technology Institute, Patras Univ., Greece Computer Technology Institute, Patras Univ., GreeceView Profile , P. Spirakis Computer Technology Institute, Patras Univ., Greece and Courant Inst. Math. Sciences, NYU Computer Technology Institute, Patras Univ., Greece and Courant Inst. Math. Sciences, NYUView Profile Authors Info & Claims SPAA '89: Proceedings of the first annual ACM symposium on Parallel algorithms and architecturesMarch 1989 Pages 254–264https://doi.org/10.1145/72935.72962Published:01 March 1989Publication History 24citation304DownloadsMetricsTotal Citations24Total Downloads304Last 12 Months3Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Given a drawing of straight lines on the plane, we wish to decide whether it is the projection of the visible part of a set of opaque polyhedra. Although there is an extensive literature and reports on empirically succesful algorithm: for this problem, there has been no definite result concerning its complexity. In this paper we show that, rather surprisingly, this problem is NP-complete. This is true even in the relatively simple case of trihedral scenes (no four planes share a point) without shadows or cracks. Despite this negative result, we present a fast algorithm for the important special case of orthohedral scenes (all planes are perpendicular to one of the three axes) with a fixed number of "possible" objects.

In [1978] Harrington and MacQueen proved that if B is an (A, E)-semirecursive subset of A, such that the functions in BA can be coded as elements of A in an (A, E)-recursive way, then ENV(A, E) is closed under the existential quantifier ∃T ∈ B. Later Moschovakis showed that if ENV(Vκ, ∈, E) is closed under the quantifier ∃t ∈ λ, where λ is the p-cofinality of κ, then the p-cofinality of κ is the least ordinal λ for which there exists a (κ, <, E)-recursive partial function ƒ into κ, such that ƒ∣λ is total from λ onto an unbounded subset of κ. In this paper we prove that for any infinite ordinal κ if p-card(κ) = κ, then ENV(κ, <, E) is closed under ∃t ∈ μ, for μ < p-cf(κ); p-cf(κ) is the “boldface” analog of p-cf((κ) and p-card(κ) is defined similarly. From this follows that for any infinite ordinal κ the following two statements are equivalent. (i) ENV(κ, <, E) is closed under bounded existential quantification. (ii) ENV(κ, <, E) = ENV(κ, <, E#) or p-cf(κ) = κ. We also show that we cannot omit any of the hypotheses in the above theorem. We follow mainly the notation of Kechris and Moschovakis [1977].