Approximating the crowd

The problem of “approximating the crowd” is that of estimating the crowd’s majority opinion by querying only a subset of it. Algorithms that approximate the crowd can intelligently stretch a limited budget for a crowdsourcing task. We present an algorithm, “CrowdSense,” that works in an online fashion where items come one at a time. CrowdSense dynamically samples subsets of the crowd based on an exploration/exploitation criterion. The algorithm produces a weighted combination of the subset’s votes that approximates the crowd’s opinion. We then introduce two variations of CrowdSense that make various distributional approximations to handle distinct crowd characteristics. In particular, the first algorithm makes a statistical independence approximation of the labelers for large crowds, whereas the second algorithm finds a lower bound on how often the current subcrowd agrees with the crowd’s majority vote. Our experiments on CrowdSense and several baselines demonstrate that we can reliably approximate the entire crowd’s vote by collecting opinions from a representative subset of the crowd.     

Approximating the max-edge-coloring problem

The max-edge-coloring problem is a natural weighted generalization of the classical edge-coloring problem arising in the domain of communication systems. In this problem each color class is assigned the weight of the heaviest edge in this class and the objective is to find a proper edge-coloring of the input graph minimizing the sum of all color classes’ weights. We present new approximation results, that improve substantially the known ones, for several variants of the problem with respect to the class of the underlying graph. In particular, we deal with variants which are known to be NP-hard (general and bipartite graphs) or are proven to be NP-hard in this paper (complete graphs with bi-valued edge weights) or whose complexity question still remains open (trees).     

Approximating traces

In order to give a semantics to concurrent processes we need a model having some good mathematical properties. To this end we generalize (infinite) Mazurkiewicz traces by adding some alphabetical information concerning the possible continuations of a process. This allows to define an approximation order compatible with the composition. We obtain a prime algebraic and coherently complete domain where the compact elements are exactly the finite approximations of processes. The composition is shown to be monotone and -continuous. We define a suitable metric which induces the Lawson topology and which yields a compact metric space being therefore complete. The finite approximations of processes form a dense and open subset and the composition is (uniformly) continuous. A preliminary version of this work appeared in [7]. Received: 2 July 1996 / 27 October 1997     

Approximating the dense set-cover problem

We study the set-cover problem, i.e. given a collection C of subsets of a finite set U, find a minimum size subset C′⊆C such that every element in U belongs to at least one member of C. An instance (C,U) of the set-cover problem is k-bounded if the number of occurrences in C of any element is bounded by a constant k?2.We present an approximation algorithm for the k-bounded set-cover problem, that achieves the ratio , where ε is defined as . If ε is relatively high, we say that the problem is dense, and this ratio in this case is better than k, which is the best known constant ratio for this problem. In the case that the number of occurrences in C of any element is exactly k=2 the problem is known as the vertex-cover problem. For dense graphs, our algorithm achieves an approximation ratio better than that of Nagamochi and Ibaraki (Japan J. Indust. Appl. Math. 16 (1999) 369), and the same approximation ratios as Karpinski and Zelikovsky (Proceedings of DIMACS Workshop on Network Design: Connectivity and Facilities Location, Vol. 40, Princeton, 1998, pp. 169-178). In our algorithm we use a combinatorial property of the set-cover problem, which is based on the classical greedy algorithm for the set-cover problem. We use this property to define a “greedy-sequence”, which is defined over a given instance of the set-cover problem and its cover.In addition, we show evidence that the ratio we achieve for the ε-dense k-bounded set-cover problem is the best constant ratio one can expect. We do this by showing that finding a better constant ratio is as hard as finding a constant ratio better than k for the k-bounded set-cover problem in which the optimal cover is known to be of size at least . (k is the best known constant ratio for this version of the k-bounded set-cover problem.) We show a similar lower bound for the approximation ratio for the vertex-cover problem in ε-dense graphs.     

On the Hardness of Approximating Spanners

A k -spanner of a connected graph G=(V,E) is a subgraph G' consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G' is larger than the distance in G by no more than a factor of k . This paper concerns the hardness of finding spanners with a number of edges close to the optimum. It is proved that for every fixed k , approximating the spanner problem is at least as hard as approximating the set-cover problem. We also consider a weighted version of the spanner problem, and prove an essential difference between the approximability of the case k=2 and the case k\geq 5 . Received October 30, 1998; revised March 4, 1999, and April 17, 2000.     

Approximating the Advertisement Placement Problem

The advertisement placement problem deals with space and time sharing by advertisements on the Internet. Consider a Web page containing a rectangular display area (e.g., a banner) in which advertisements may appear. The display area can be utilized efficiently by allowing several small ads to appear simultaneously side by side, as well as by cycling through a schedule of ads, allowing different ads to be displayed at different times. A customer wishing to purchase advertising space specifies an ad size and a display count, which is the number of times their ad should appear during each cycle. The scheduler may accept or reject any given advertisement, but must be able to schedule all accepted ads within the given time and space constraints. Each advertisement has a non-negative profit associated with it, and the objective is to schedule a maximum-profit subset of ads. We present a (3 + )-approximation algorithm for the general problem, as well as (2 + )-approximation algorithms for two special cases.     

Approximating Geographical Queries

This article proposes a graph-theoretic methodology for query approximation in Geographic Information Systems, enabling the relaxation of three kinds of query constraints: topological, semantic and structural. An approximate query is associated with a value corresponding to the degree of similarity with the original query. Such a value is computed for topological constraints on the basis of the topological distance between configurations, for semantic constraints using the information content approach, and for structural constraints revisiting the maximum weighted matching problem in bipartite graphs. Finally, the high correlation of our proposal with human judgment is demonstrated by an experiment.     

Approximating the Minimum Chain Completion problem

A bipartite graph G=(U,V,E) is a chain graph [M. Yannakakis, Computing the minimum fill-in is NP-complete, SIAM J. Algebraic Discrete Methods 2 (1) (1981) 77–79] if there is a bijection such that Γ(π(1))Γ(π(2))Γ(π(|U|)), where Γ is a function that maps a node to its neighbors.We give approximation algorithms for two variants of the Minimum Chain Completion problem, where we are given a bipartite graph G(U,V,E), and the goal is find the minimum set of edges F that need to be added to G such that the bipartite graph G^′=(U,V,E^′) (E^′=EF) is a chain graph.     

Approximating minimum cocolorings

A cocoloring of a graph G is a partition of the vertex set of G such that each set of the partition is either a clique or an independent set in G. Some special cases of the minimum cocoloring problem are of particular interest.We provide polynomial-time algorithms to approximate a minimum cocoloring on graphs, partially ordered sets and sequences. In particular, we obtain an efficient algorithm to approximate within a factor of 1.71 a minimum partition of a partially ordered set into chains and antichains, and a minimum partition of a sequence into increasing and decreasing subsequences.     

Approximating Latin Square Extensions

In this paper we investigate the problem of computing the maximum number of entries which can be added to a partially filled latin square. The decision version of this question is known to be NP-complete. We present two approximation algorithms for the optimization version of this question. We first prove that the greedy algorithm achieves a factor of 1/3. We then use insights derived from the linear relaxation of an integer program to obtain an algorithm based on matchings that achieves a better performance guarantee of 1/2. These are the first known polynomial-time approximation algorithms for the latin square completion problem that achieve nontrivial worst-case performance guarantees. Our study is motivated by applications to lightpath assignment and switch configuration in wavelength routed multihop optical networks. Received September 25, 1997; revised February 16, 1998.     

Approximating Biconnectivity in Parallel

Consider the following NP-hard problems: Given a graph G , find minimum 2-edge connected and 2-vertex connected subgraphs spanning all vertices of G . Over the past few years, exciting sequential algorithms for approximating such minimum subgraphs have been produced [6],[10]. The approximation factors are improved from 2 to 3/2 for both of the problems. Yet the techniques involved are all based on augmenting depth-first-search trees and no similar progress has been carried to the parallel context. This paper presents NC algorithms to achieve approximation factors of 3/2 + ε for these two problems without computing depth-first-search trees. Received June 21, 1995; revised December 24, 1996, and June 2, 1997.     

Approximating Satisfiable Satisfiability Problems

We study the approximability of the Maximum Satisfiability Problem (MAX SAT) and of the boolean k -ary Constraint Satisfaction Problem (MAX k CSP) restricted to satisfiable instances. For both problems we improve on the performance ratios of known algorithms for the unrestricted case. Our approximation for satisfiable MAX 3CSP instances is better than any possible approximation for the unrestricted version of the problem (unless P=NP). This result implies that the requirement of perfect completeness weakens the acceptance power of non-adaptive PCP verifiers that read 3 bits. We also present the first non-trivial results about PCP classes defined in terms of free bits that collapse to P. Received August 1997; revised February 1999.     

Approximating multi-objective scheduling problems

In many practical situations, decisions are multi-objective by nature. In this paper, we propose a generic approach to deal with multi-objective scheduling problems (MOSPs). The aim is to determine the set of Pareto solutions that represent the interactions between the different objectives. Due to the complexity of MOSPs, an efficient approximation based on dynamic programming is developed. The approximation has a provable worst case performance guarantee. Even though the approximate Pareto set consists of fewer solutions, it represents a good coverage of the true set of Pareto solutions. We consider generic cost parameters that depend on the state of the system. Numerical results are presented for the time-dependent multi-objective knapsack problem, showing the value of the approximation in the special case when the state of the system is expressed in terms of time.     

Approximating geodesic tree distance

Billera, Holmes, and Vogtmann introduced an intriguing new phylogenetic tree metric for weighted trees with useful properties related to statistical analysis. However, the best known algorithm for calculating this distance is exponential in the number of leaves of the trees compared. We point out that lower and upper bounds for this distance, which can be calculated in linear time, can differ by at most a multiplicative factor of .     

Approximating Node-Connectivity Augmentation Problems

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J=(V,E _J ) and connectivity requirements $\{r(u,v): u,v \in V\}$ , find a minimum size set I of new edges (any edge is allowed) such that the graph J∪I contains r(u,v) internally-disjoint uv-paths, for all u,v∈V. In Rooted NCA there is s∈V such that r(u,v)>0 implies u=s or v=s. For large values of k=max? _u,v∈V r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit an approximation ratio polylogarithmic in k. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(kln?n) for NCA and O(ln?n) for Rooted NCA. In this paper we give an approximation algorithm with ratios O(kln?² k) for NCA and O(ln?² k) for Rooted NCA. This is the first approximation algorithm with ratio independent of?n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If $\mathcal{D}$ is a set of node pairs in a graph?J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in $\mathcal{D}$ is O(? ²), where ? is the maximum connectivity in J of a pair in $\mathcal{D}$ .