Anke van Zuylen

Deterministic Pivoting Algorithms for Constrained Ranking and Clustering Problems

We introduce new problems of finding minimum-cost rankings and clusterings which must be consistent with certain constraints (e.g. an input partial order in the case of ranking problems); we give deterministic approximation algorithms for these problems. Randomized approximation algorithms for unconstrained versions of these problems were given by Ailon, Charikar, and Newman [2] and by Ailon and Charikar [1]. Finding deterministic approximation algorithms for these problems answers an open question of Ailon et al. [2].n In particular, we give deterministic algorithms for constrained weighted feedback arc set in tournaments, constrained correlation clustering, and constrained hierarchical clustering related to finding good ultrametrics. Our algorithms follow the paradigm of Ailon et al. [2] of choosing a particular vertex as a pivot and partitioning the graph according to the pivot; unlike their algorithms, we do not choose the pivot randomly but rather use an LP relaxation to choose a good pivot deterministically. Additionally, the use of the LP relaxation allows us to impose constraints easily and analyze the results. In several cases we are able to find approximation factors for the constrained problems that improve on the factors they obtained for the unconstrained cases. We also give a combinatorial algorithm for constrained weighted feedback arc set in tournaments with weights satisfying probability constraints. This algorithm improves on the best known factor given by deterministic combinatorial algorithms for the unconstrained case.

Simpler Approximation of the Maximum Asymmetric Traveling Salesman Problem

We give a very simple approximation algorithm for the maximum asymmetric traveling salesman problem. The approximation guarantee of our algorithm is 2/3, which matches the best known approximation guarantee by Kaplan, Lewenstein, Shafrir and Sviridenko. Our algorithm is simple to analyze, and contrary to previous approaches, which need an optimal solution to a linear program, our algorithm is combinatorial and only uses maximum weight perfect matching algorithm.

Improved Approximation Algorithms for Bipartite Correlation Clustering

In this work we study the problem of bipartite correlation clustering (BCC), a natural bipartite counterpart of the well-studied correlation clustering (CC) problem [N. Bansal, A. Blum, and S. Chawla, Machine Learning, 56 (2004), pp. 89--113], also referred to as graph editing [R. Shamir, R. Sharan, and D. Tsur, Discrete Appl. Math., 144 (2004), pp. 173--182]. Given a bipartite graph, the objective of BCC is to generate a set of vertex disjoint bicliques (clusters) that minimizes the symmetric difference to the original graph. The best-known approximation algorithm for BCC due to Amit [N. Amit, The Bicluster Graph Editing Problem, Masters Thesis, Tel Aviv University, Tel Aviv, Israel, 2004] guarantees an

Linear programming based approximation algorithms for feedback set problems in bipartite tournaments

11

Simpler 3/4-approximation algorithms for MAX SAT

-approximation ratio. In this paper we present two algorithms. The first is a linear program based

Linear Programming Based Approximation Algorithms for Feedback Set Problems in Bipartite Tournaments

4

Greedy Algorithms for the Maximum Satisfiability Problem: Simple Algorithms and Inapproximability Bounds

-approximation algorithm. Like the previous approximation algorithm, it requires solving a large convex problem, which becomes prohibitive even for modestly sized tasks. The second algorithm, and our...

On the integrality gap of the subtour LP for the 1,2-TSP

We consider the feedback vertex set and feedback arc set problems on bipartite tournaments. We improve on recent results by giving a 2-approximation algorithm for the feedback vertex set problem. We show that this result is the best that we can attain when using optimal solutions to a certain linear program as a lower bound on the optimal value. For the feedback arc set problem on bipartite tournaments, we show that a recent 4-approximation algorithm proposed by Gupta (2008) [8] is incorrect. We give an alternative 4-approximation algorithm based on an algorithm for the feedback arc set on (non-bipartite) tournaments given by van Zuylen and Williamson (2009) [14].

On Some Recent Approximation Algorithms for MAX SAT

We consider the recent randomized 3/4-algorithm for MAX SAT of Poloczek and Schnitger. We give a much simpler set of probabilities for setting the variables to true or false, which achieve the same expected performance guarantee. Our algorithm suggests a conceptually simple way to get a deterministic algorithm: rather than comparing to an unknown optimal solution, we instead compare the algorithms output to the optimal solution of an LP relaxation. This gives rise to a new LP rounding algorithm, which also achieves a performance guarantee of 3/4.

Multiplying pessimistic estimators: deterministic approximation of max TSP and maximum triangle packing

We consider the feedback vertex set and feedback arc set problems in bipartite tournaments. We improve on recent results by giving a 2-approximation algorithm for the feedback vertex set problem. We show that this result is the best we can attain when using a certain linear program as the lower bound on the optimal value. For the feedback arc set problem in bipartite tournaments, we show that a recent 4-approximation algorithm proposed by Gupta [5,6] is incorrect. We give an alternative 4-approximation algorithm based on an algorithm for feedback arc set in (regular) tournaments in [10,11].