Sampling for Beyond-Worst-Case Online Ranking

The minimum feedback arc set problem is an NP-hard problem on graphs that seeks a minimum set of arcs which, when removed from the graph, leave it acyclic. In this work, we investigate several approximations for computing a minimum feedback arc set with the goal of comparing the quality of the solutions and the running times. Our investigation is motivated by applications in Social Network Analysis such as misinformation removal and label propagation. We present careful algorithmic engineering for multiple algorithms to improve the scalability of each approach. In particular, two approaches we optimize (one greedy and one randomized) provide a nice balance between feedback arc set size and running time complexity. We experimentally compare the performance of a wide range of algorithms on a broad selection of large online networks including Twitter, LiveJournal, and the Clueweb12 dataset. The experiments reveal that our greedy and randomized implementations outperform the other approaches by simultaneously computing a feedback arc set of competitive size and scaling to web-scale graphs with billions of vertices and tens of billions of arcs. Finally, we extend the algorithms considered to the probabilistic case in which arcs are realized with some fixed probability and provide detailed experimental comparisons.

A tournament T = (V , A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed ϵ > 0, the kernelized instance has at most (2 + ϵ)k vertices. Our result improves the previous known bound of O(k2 ) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k- FAST.

Feedback arc set in bipartite tournaments is NP-complete

SummaryAsian options are popular path-dependent derivatives in the financial market. However, how to price them efficiently and accurately has been a longstanding research and practical problem. No known exact pricing formulas are available to price the Asian option. Although approximate pricing formulas exist, they lack accuracy guarantees. Asian options can be priced on the lattice. A lattice divides a time interval into n equal-length time steps. It is known that the value computed by the lattice converges to the true option value as n → ∞. Unfortunately, only subexponential-time algorithms are available if Asian options are to be priced on the lattice without approximations. Efficient approximation algorithms are available for the lattice. The best known in the literature is an O(n 3.5)-time approximation lattice algorithm and an O(n 3)-time approximation PDE algorithm. Our paper suggests an O(n 2.5)-time lattice algorithm. Our algorithm uses a novel technique based on the method of Lagrange multipliers to minimize the approximation error. Numerical results verify the accuracy and the excellent performance of our algorithm.

The data analysis problem of ordering or sequencing a set of objects using an asymmetric proximity function is reviewed, with an emphasis on literature not generally referenced in psychology. In particular, an attempt is made to present some of the more important approaches to the asymmetric seriation task that have been developed independently under a number of different names, for instance, the triangularization of input‐output matrices, minimum feedback arc sets, maximum likelihood paired comparison ranking, majority rule under transitivity constraints, and so on. A simple numerical example that illustrates some of the basic concepts is presented in the course of the discussion.

We present a O(|A|(|V | + |A|)) heuristic Algorithm for the minimum weighted feedback arc set problem.The Algorithm is based on two linear arrangements of the set of arcs, and on the concept of minimal and stable feedback arc set previously introduced and now generalized to the weighted case.The obtained results, compared to two other state of the art Algorithms, show a high level of efficacy of the proposed Algorithm.

Memory management is a fundamental problem in computer architecture and operating systems. We consider a two-level memory system with fast, but small cache and slow, but large main memory. The underlying theoretical problem is known as the paging problem. A sequence of requests to pages has to be served by making each requested page available in the cache. A paging strategy replaces pages in the cache with requested ones. The aim is to minimize the number of page faults that occur whenever a requested page is not in the cache.Experience shows that the paging strategy LEAST-RECENTLY-USED (LRU) usually achieves a factor around 2 to 3 compared to the optimum number of faults. This contrasts the theoretical worst case, in which this factor can be as large as the cache size k.One difficulty in analyzing the paging problem was the lack of an appropriate lower bound for the minimum number of page faults. We address this issue and propose a general lower bound which provides insight into the global structure of a given request sequence. In addition, we derive a characterization for the number of faults incurred by LRU.We give a theoretical explanation why LRU performs well in practice. We classify the set of all request sequences according to certain parameters and prove a bound on the competitive ratio of LRU, which depends on them. This bound varies between 2 and k, i.e., it includes the worst-case, but explains for which sequences LRU achieves constant competitive ratio. The classification is motivated from the structure of request sequences of practical applications: locality of reference and characteristic data access patterns. We argue that this structure yields values around 2 for our bound. Indeed, it is between 2 and 5 in extensive practical experiments.Furthermore, we study the paging problem with variable cache size, which was already considered previously. We show that this approach is not appropriate to explain the usual good performance of LRU. We measure the performance of LRU with the expected competitive ratio E[ALG]/E[OPT] and the expected performance ratio E[ALG]/E[OPT] in a diffuse adversary model and compare both measures. Our analysis yields that the expected competitive ratio gives a misleading answer.

In this paper, the authors consider an on-line scheduling problem of m (m ≥ 3) identical machines with common maintenance time interval and nonresumable availability. For the case that the length of maintenance time interval is larger than the largest processing time of jobs, the authors prove that any on-line algorithm has not a constant competitive ratio. For the case that the length of maintenance time interval is less than or equal to the largest processing time of jobs, the authors prove a lower bound of 3 on the competitive ratio. The authors give an on-line algorithm with competitive ratio \(4 - \tfrac{1} {m} \). In particular, for the case of m = 3, the authors prove the competitive ratio of the on-line algorithm is \(\tfrac{{10}} {3} \).

On approximating the rank of graph divisors

The feedback arc set of a directed graph is the subset of the graph that consists of at least one edge from every cycle in the graph. Removing the fewest number of these edges gives the minimum feedback arc set. Three methods that attempt to solve this problem use equivalency and NP reductions, an approximate greedy algorithm, and an integer linear program for an exact solution. The minimum FAS can be applied to real world events such as sporting tournaments and ranked voting.

Constructing a tree spanning a given subset of vertices of a graph is a generic problem that has been extensively studied (for example the classical Steiner tree is one of them). In this paper, we propose to take into account a problem that has not been often studied: the dynamic part, in which new vertices must be inserted in a given tree. We propose to do these insertions with respect of structural constraints that we describe and to capture the efficiency of algorithms by a particular type of competitive ratios. The parameter to optimize in our study is related to distances between the connected vertices. Our main results are the following. We propose a method, called median-addition. We prove that its competitive ratio is non constant but optimal. This result is proved under the hypothesis that the algorithm does not know the sequence of added vertices (called unknown-future or online model). This method can be used to design communication structures for applications like video-conference in networks for example. In another part of the paper, we prove the surprising following result: even if the algorithm knows the sequence of additions of new vertices (hence everything is known in advance), no algorithm can have a constant competitive ratio.

Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

Feedback arc set problem in bipartite tournaments

Extracting interesting points from a large dataset is an important problem for multi-criteria decision making. Recently, k-regret query was proposed and received attentions from the database community because it does not require any utility function from users and the output size is controllable. In this paper, we consider k-regret queries with binary constraints which doesn’t be addressed before. Given a collection of binary constraints, we study the problem of extracting the k representative points with small regret ratio while satisfying the binary constraints. To express the satisfaction of data points by the binary constraints, we propose two models named NBC and LBC in quantitative and qualitative ways respectively. In quantitative way, the satisfaction is expressed by a real number between 0 and 1 which quaNtifies the satisfaction of a point by the Binary Constraints. While in qualitative way, the satisfaction is modeled quaLitatively by a set of Binary Constraints which are satisfied by the point. Further, two efficient approximate algorithms called \(NBC_P\)-Greedy and \(NBC_{DN}\)-Greedy are developed based on NBC while \(LBC_{DN}\)-Greedy algorithm is proposed based on LBC. Extensive experiments on synthetic and real datasets confirm the efficiency and effectiveness of our proposed algorithms.

A sequence of objects which are characterized by their color has to be processed. Their processing order influences how efficiently they can be processed: Each color change between two consecutive objects produces costs. A reordering buffer which is a random access buffer with storage capacity for k objects can be used to rearrange this sequence online in such a way that the total costs are reduced. This concept is useful for many applications in computer science and economics. The strategy with the best known competitive ratio is MAP. An upper bound of O(log k) on the competitive ratio of MAP is known and a non-constant lower bound on the competitive ratio is not known [2]. Based on theoretical considerations and experimental evaluations, we give strong evidence that the previously used proof techniques are not suitable to show an $o(\sqrt{\log k})$ upper bound on the competitive ratio of MAP. However, we also give some evidence that in fact MAP achieves a competitive ratio of O(1). Further, we evaluate the performance of several strategies on random input sequences experimentally. MAP and its variants RC and RR clearly outperform the other strategies FIFO, LRU, and MCF . In particular, MAP, RC, and RR are the only known strategies whose competitive ratios do not depend on the buffer size. Furthermore, MAP achieves the smallest constant competitive ratio.