A Theoretical Analysis of Query Selection for Collaborative Filtering

We consider the problem of determining which of a set of experts has tastes most similar to a given user by asking the user questions about his likes and dislikes. We describe a simple algorithm for generating queries for a theoretical model of this problem. We show that the algorithm requires at most opt(F)(ln(|F|/opt(F)) + 1) + 1 queries to find the correct expert, where opt(F) is the optimal worst-case bound on the number of queries for learning arbitrary elements of the set of experts F. The algorithm runs in time polynomial in |F| and |X| (where X is the domain) and we prove that no polynomial-time algorithm can have a significantly better bound on the number of queries unless all problems in NP have n ^{O(log log n)} time algorithms. We also study a more general case where the user ratings come from a finite set Y and there is an integer-valued loss function on Y that is used to measure the distance between the ratings. Assuming that the loss function is a metric and that there is an expert within a distance from the user, we give a polynomial-time algorithm that is guaranteed to find such an expert after at most 2opt(F, ) ln + 2( + 1)(1 + deg(F, )) queries, where deg(F, ) is the largest number of experts in F that are within a distance 2 of any f F.     

一个正则NP-完全问题及其不可近似性

通过一个适当的归约变换,可以将一个CNF (conjunctive normal form)公式变换为另一个具有某种特殊结构或性质的公式,使两者具有相同的可满足性.带有正则结构的CNF公式的因子图在图论中具有某些良好的性质和结果,可以用于研究公式的可满足性和计算复杂性.极小不可满足公式具有一个临界特征,公式本身不可满足,从原始公式中删去任意一个子句后得到的公式可满足.借助此临界特性,给出了一个从3-CNF公式到正则(3,4)-CNF公式的多项式归约转换.这里,正则(3,4)-CNF公式是指公式中每个子句的长度恰为3,每个变元出现的次数恰为4.因此,正则(3,4)-SAT问题是一个NP-完全问题,并且MAX(3,4)-SAT是不可近似问题.     

Progress in Computational Complexity Theory

We briefly survey a number of important recent achievements in Theoretical Computer Science (TCS), especially Computational Complexity Theory. We will discuss the PCP Theorem, its implications to inapproximability on combinatorial optimization problems; space bounded computations, especially deterministic logspace algorithm for undirected graph connectivity problem; deterministic polynomial-time primality test; lattice complexity, worst-case to average-case reductions; pseudorandomness and extractor constructions; and Valiant's new theory of holographic algorithms and reductions.     

NP-Hardness of Broadcast Scheduling and Inapproximability of Single-Source Unsplittable Min-Cost Flow

We consider the version of broadcast scheduling where a server can transmit W messages of a given set at each time-step, answering previously made requests for these messages. The goal is to minimize the average response time (ART) if the amount of requests is known in advance for each time-step and message. We prove that this problem is NP-hard, thus answering an open question stated by Kalyanasundaram, Pruhs and Velauthapillai (Proceedings of ESA 2000, LNCS 1879, 2000, pp. 290–301). Furthermore, we present an approximation algorithm that is allowed to send several messages at once. Using six channels for transmissions, the algorithm achieves an ART that is at least as good as the optimal solution using one channel. From the NP-hardness of broadcast scheduling we derive a new inapproximability result of (2 − ε, 1) for the (congestion, cost) bicriteria version of the single source unsplittable min-cost flow problem, for arbitrary ε > 0. The result holds even in the often considered case where the maximum demand is less than or equal to the minimum edge capacity (d _max ≤ u _min), a case for which an algorithm with ratio (3, 1) was presented by Skutella.     

近似算法之测度视角

标准近似、微分近似和占优分析是三种不同的近似算法度量方法。标准近似比度量近似解偏离最优解的相对误差。微分近似关注近似解解值在最优解值和最差解值所形成的区间内所处的位置。占优分析考虑近似解在所有可行解中的排名。本文综述相关概念和主要成果,以及各测度方法的优缺点。尤其关注以PCP定理及唯一博弈猜想为代表的不可近似性成果。     

Two dependency constrained spanning tree problems

We introduce two spanning tree problems with dependency constraints, where an edge can be chosen only if at least one or all edges in its dependency set are also chosen, respectively. The dependencies on the input graph G are described by a digraph D whose vertices are the edges of G, and the in‐neighbors of a vertex are its dependency set. We show that both problems are NP‐hard even if G is an outerplanar chordal graph with diameter 2 or maximum degree 3, and D is a disjoint union of arborescences of height 2. We also prove that the problems are inapproximable to a factor, unless , and that they are W[2]‐hard. As an attempt to narrow the hardness gap, we present two polynomial cases. We test integer linear programming formulations based on directed cut (DCUT) and Miller–Tucker–Zemlin (MTZ) constraints. Computational experiments are reported.