No Arabic abstract
Constraint satisfaction problems (CSPs) models many important intractable NP-hard problems such as propositional satisfiability problem (SAT). Algorithms with non-trivial upper bounds on running time for restricted SAT with bounded clause length k (k-SAT) can be classified into three styles: DPLL-like, PPSZ-like and Local Search, with local search algorithms having already been generalized to CSP with bounded constraint arity k (k-CSP). We generalize a DPLL-like algorithm in its simplest form and a PPSZ-like algorithm from k-SAT to k-CSP. As far as we know, this is the first attempt to use PPSZ-like strategy to solve k-CSP, and before little work has been focused on the DPLL-like or PPSZ-like strategies for k-CSP.
We consider the generalized $k$-server problem on uniform metrics. We study the power of memoryless algorithms and show tight bounds of $Theta(k!)$ on their competitive ratio. In particular we show that the textit{Harmonic Algorithm} achieves this competitive ratio and provide matching lower bounds. This improves the $approx 2^{2^k}$ doubly-exponential bound of Chiplunkar and Vishwanathan for the more general setting of uniform metrics with different weights.
Given a metric $(V,d)$ and a $textsf{root} in V$, the classic $textsf{$k$-TSP}$ problem is to find a tour originating at the $textsf{root}$ of minimum length that visits at least $k$ nodes in $V$. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochast
We extend the notion of a strong backdoor from the CSP setting to the Valued CSP setting (VCSP, for short). This provides a means for augmenting a class of tractable VCSP instances to instances that are outside the class but of small distance to the class, where the distance is measured in terms of the size of a smallest backdoor. We establish that VCSP is fixed-parameter tractable when parameterized by the size of a smallest backdoor into every tractable class of VCSP instances characterized by a (possibly infinite) tractable valued constraint language of finite arity and finite domain. We further extend this fixed-parameter tractability result to so-called scattered classes of VCSP instances where each connected component may belong to a different tractable class.
A heuristic model procedure for determining satisfiability of CNF-formulae is set up and described by nonlinear recursion relations for m (number of clauses), n (number of variables) and clause filling k. The system mimicked by the recursion undergoes a sharp transition from bounded running times (easy) to uncontrolled runaway behaviour (hard). Thus the parameter space turns out to be separated into regions with qualitatively different efficiency of the model procedure. The transition results from a competition of exponential blow up by branching versus growing number of orthogonal clauses.
We study fair clustering problems as proposed by Chierichetti et al. (NIPS 2017). Here, points have a sensitive attribute and all clusters in the solution are required to be balanced with respect to it (to counteract any form of data-inherent bias). Previous algorithms for fair clustering do not scale well. We show how to model and compute so-called coresets for fair clustering problems, which can be used to significantly reduce the input data size. We prove that the coresets are composable and show how to compute them in a streaming setting. Furthermore, we propose a variant of Lloyds algorithm that computes fair clusterings and extend it to a fair k-means++ clustering algorithm. We implement these algorithms and provide empirical evidence that the combination of our approximation algorithms and the coreset construction yields a scalable algorithm for fair k-means clustering.