Реоптимізація проблем про узагальнену виконуваність з предикатами розмірності 2

Припустимо, що виконується унікальна ігрова гіпотеза (UGC). Тоді для реоптимізації Max Cut (при добавленні довільного ребра) існує поліноміальний пороговий (оптимальний) φ(αGW)-наближений алгоритм, де φ(αGW)=1/(2−αGW)≈0,891716, при цьому αGW≈0,878567 (константа Гоеманса–Уільямсона). Для реоптимізації Max 2-Sat (при добавленні довільної диз'юнкції) існує поліноміальний пороговий (оптимальний) φ(α^−LLZ)-наближений алгоритм, де φ(α^−LLZ)≈0,943544, при цьому α^−LLZ≈0,940166 (константа Левіна–Лівната–Звіка).Допустим, что выполняется уникальная игровая гипотеза (UGC). Тогда для реоптимизации Max Cut (при вставке произвольного ребра) существует полиномиальный пороговый (оптимальный) φ(αGW)-приближенный алгоритм, где φ(αGW)=1/(2−αGW)≈0,891716, при этом αGW≈0,878567 (константа Гоеманса–Уильямсона). Для реоптимизации Max 2-Sat (при вставке произвольной дизьюнкции) существует полиномиальный пороговый (оптимальный) φ(α^−LLZ)-приближенный алгоритм, где φ(α^−LLZ)≈0,943544, при этом α^−LLZ≈0,940166 (константа Левина–Ливната–Звика).Assume that the Unique Game Conjecture (UGC) is held. Then, for the reoptimization of Max Cut (if a new edge is inserted), a polynomial threshold (optimal) φ(αGW)-approximation algorithm exists, where φ(αGW)=1/(2−αGW)≈0.891716 and αGW≈0.878567 (the Goemans–Williamson constant). For the reoptimization of Max 2-Sat (if a new disjunction is inserted), a polynomial threshold (optimal) φ(α^−LLZ)-approximation algorithm exists, where φ(α^−LLZ)≈0.943544 and α^−LLZ≈0.940166 (the Levin–Livnat–Zwick constant)

Approximate Graph Coloring by Semidefinite Programming

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)} log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz theta-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the theta-function

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

A Logical Approach to Efficient Max-SAT solving

Weighted Max-SAT is the optimization version of SAT and many important problems can be naturally encoded as such. Solving weighted Max-SAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in finding efficient solving techniques. Most of this work focus on the computation of good quality lower bounds to be used within a branch and bound DPLL-like algorithm. Most often, these lower bounds are described in a procedural way. Because of that, it is difficult to realize the {\em logic} that is behind. In this paper we introduce an original framework for Max-SAT that stresses the parallelism with classical SAT. Then, we extend the two basic SAT solving techniques: {\em search} and {\em inference}. We show that many algorithmic {\em tricks} used in state-of-the-art Max-SAT solvers are easily expressable in {\em logic} terms with our framework in a unified manner. Besides, we introduce an original search algorithm that performs a restricted amount of {\em weighted resolution} at each visited node. We empirically compare our algorithm with a variety of solving alternatives on several benchmarks. Our experiments, which constitute to the best of our knowledge the most comprehensive Max-sat evaluation ever reported, show that our algorithm is generally orders of magnitude faster than any competitor