Binarisation for Valued Constraint Satisfaction Problems

We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. Second, we extend the reduction of CSPs to binary CSPs described by Bul´ın et al. [Log. Methods Comput. Sci., 11 (2015)] to VCSPs. This reduction establishes that VCSPs over a fixed valued constraint language are polynomial-time equivalent to minimum-cost homomorphism problems over a fixed digraph

Achieving Long-term Fairness in Submodular Maximization through Randomization

Submodular function optimization has numerous applications in machine learning and data analysis, including data summarization which aims to identify a concise and diverse set of data points from a large dataset. It is important to implement fairness-aware algorithms when dealing with data items that may contain sensitive attributes like race or gender, to prevent biases that could lead to unequal representation of different groups. With this in mind, we investigate the problem of maximizing a monotone submodular function while meeting group fairness constraints. Unlike previous studies in this area, we allow for randomized solutions, with the objective being to calculate a distribution over feasible sets such that the expected number of items selected from each group is subject to constraints in the form of upper and lower thresholds, ensuring that the representation of each group remains balanced in the long term. Here a set is considered feasible if its size does not exceed a constant value of $b$. Our research includes the development of a series of approximation algorithms for this problem.Comment: This paper has been accepted to 19th Cologne-Twente Workshop on Graphs and Combinatorial Optimizatio

Adapting Local Sequential Algorithms to the Distributed Setting

It is a well known fact that sequential algorithms which exhibit a strong "local" nature can be adapted to the distributed setting given a legal graph coloring. The running time of the distributed algorithm will then be at least the number of colors. Surprisingly, this well known idea was never formally stated as a unified framework. In this paper we aim to define a robust family of local sequential algorithms which can be easily adapted to the distributed setting. We then develop new tools to further enhance these algorithms, achieving state of the art results for fundamental problems. We define a simple class of greedy-like algorithms which we call orderless-local algorithms. We show that given a legal c-coloring of the graph, every algorithm in this family can be converted into a distributed algorithm running in O(c) communication rounds in the CONGEST model. We show that this family is indeed robust as both the method of conditional expectations and the unconstrained submodular maximization algorithm of Buchbinder et al. [Niv Buchbinder et al., 2015] can be expressed as orderless-local algorithms for local utility functions - Utility functions which have a strong local nature to them. We use the above algorithms as a base for new distributed approximation algorithms for the weighted variants of some fundamental problems: Max k-Cut, Max-DiCut, Max 2-SAT and correlation clustering. We develop algorithms which have the same approximation guarantees as their sequential counterparts, up to a constant additive epsilon factor, while achieving an O(log^* n) running time for deterministic algorithms and O(epsilon^{-1}) running time for randomized ones. This improves exponentially upon the currently best known algorithms

Approximation algorithms for clustering and facility location problems

In this thesis we design and analyze algorithms for various facility location and clustering problems. The problems we study are NP-Hard and therefore, assuming P is not equal NP, there do not exist polynomial time algorithms to solve them optimally. One approach to cope with the intractability of these problems is to design approximation algorithms which run in polynomial-time and output a near-optimal solution for all instances of the problem. However these algorithms do not always work well in practice. Often heuristics with no explicit approximation guarantee perform quite well. To bridge this gap between theory and practice, and to design algorithms that are tuned for instances arising in practice, there is an increasing emphasis on beyond worst-case analysis. In this thesis we consider both these approaches. In the first part we design worst case approximation algorithms for Uniform Submodular Facility Location (USFL), and Capacitated k-center (CapKCenter) problems. USFL is a generalization of the well-known Uncapacitated Facility Location problem. In USFL the cost of opening a facility is a submodular function of the clients assigned to it (the function is identical for all facilities). We show that a natural greedy algorithm (which gives constant factor approximation for Uncapacitated Facility Location and other facility location problems) has a lower bound of log(n), where n is the number of clients. We present an O(log^2 k) approximation algorithm where k is the number of facilities. The algorithm is based on rounding a convex relaxation. We further consider several special cases of the problem and give improved approximation bounds for them. The CapKCenter problem is an extension of the well-known k-center problem: each facility has a maximum capacity on the number of clients that can be assigned to it. We obtain a 9-approximation for this problem via a linear programming (LP) rounding procedure. Our result, combined with previously known lower bounds, almost settles the integrality gap for a natural LP relaxation. In the second part we consider several well-known clustering problems like k-center, k-median, k-means and their corresponding outlier variants. We use beyond worst-case analysis due to the practical relevance of these problems. In particular we show that when the input instances are 2-perturbation resilient (i.e. the optimal solution does not change when the distances change by a multiplicative factor of 2), the LP integrality gap for k-center (and also asymmetric k-center) is 1. We further introduce a model of perturbation resilience for clustering with outliers. Under this new model, we show that previous results (including our LP integrality result) known for clustering under perturbation resilience also extend for clustering with outliers. This leads to a dynamic programming based heuristic for k-means with outliers (k-means-outlier) which gives an optimal solution when the instance is 2-perturbation resilient. We propose two more algorithms for k-means-outlier — a sampling based algorithm which gives an O(1) approximation when the optimal clusters are not “too small”, and an LP rounding algorithm which gives an O(1) approximation at the expense of violating the number of clusters and outliers by a small constant. We empirically study our proposed algorithms on several clustering datasets

Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game

In the past 20 years, increasing complexity in real world data has lead to the study of higher-order data models based on partitioning hypergraphs. However, hypergraph partitioning admits multiple formulations as hyperedges can be cut in multiple ways. Building upon a class of hypergraph partitioning problems introduced by Li & Milenkovic, we study the problem of minimizing ratio-cut objectives over hypergraphs given by a new class of cut functions, monotone submodular cut functions (mscf's), which captures hypergraph expansion and conductance as special cases. We first define the ratio-cut improvement problem, a family of local relaxations of the minimum ratio-cut problem. This problem is a natural extension of the Andersen & Lang cut improvement problem to the hypergraph setting. We demonstrate the existence of efficient algorithms for approximately solving this problem. These algorithms run in almost-linear time for the case of hypergraph expansion, and when the hypergraph rank is at most $O(1)$. Next, we provide an efficient $O(\log n)$-approximation algorithm for finding the minimum ratio-cut of $G$. We generalize the cut-matching game framework of Khandekar et. al. to allow for the cut player to play unbalanced cuts, and matching player to route approximate single-commodity flows. Using this framework, we bootstrap our algorithms for the ratio-cut improvement problem to obtain approximation algorithms for minimum ratio-cut problem for all mscf's. This also yields the first almost-linear time $O(\log n)$-approximation algorithms for hypergraph expansion, and constant hypergraph rank. Finally, we extend a result of Louis & Makarychev to a broader set of objective functions by giving a polynomial time $O\big(\sqrt{\log n}\big)$-approximation algorithm for the minimum ratio-cut problem based on rounding $\ell_2^2$-metric embeddings.Comment: Comments and feedback welcom