强基可序拟阵上加权拟阵匹配的一个简单PTAS

An EPTAS for Budgeted Matching and Budgeted Matroid Intersection

We study the budgeted versions of the well known matching and matroid intersection problems. While both problems admit a polynomial-time approximation scheme (PTAS) [Berger et al. (Math. Programming, 2011), Chekuri, Vondrak and Zenklusen (SODA 2011)], it has been an intriguing open question whether these problems admit a fully PTAS (FPTAS), or even an efficient PTAS (EPTAS). In this paper we answer the second part of this question affirmatively, by presenting an EPTAS for budgeted matching and budgeted matroid intersection. A main component of our scheme is a novel construction of representati…

带线性约束的拟阵优化问题的下界

Lower Bounds for Matroid Optimization Problems with a Linear Constraint

本文证明带线性约束的拟阵优化问题族（含预算拟阵独立集）均无完全多项式时间近似方案，通过引入Π-拟阵族证明精确权重拟阵基无伪多项式算法，解决了其复杂性开放问题。

An EPTAS for Budgeted Matroid Independent Set

We consider the budgeted matroid independent set problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a matroid over the elements and a budget. The goal is to select a subset of elements which maximizes the total profit subject to the matroid and budget constraints. Several well known special cases, where we have, e.g., a uniform matroid and a budget, or no matroid constraint (i.e., the classic knapsack problem), admit a fully polynomial-time approximation scheme (FPTAS). In contrast, already a slight generalization to the multi-budgeted matro…

Optimal Matroid Partitioning Problems

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given a finite set $E$ and $k$ weighted matroids $(E, \mathcal{I}_i, w_i)$, $i = 1, \dots, k$, and our task is to find a minimum partition $(I_1,\dots,I_k)$ of $E$ such that $I_i \in \mathcal{I}_i$ for all $i$. For each objective function, we give a polynomial-time algorithm or prove NP-hardness. In particular, for the case when the given weighted matroids are identical and the objective function is the sum of the maximum weight in each set (i.e., $\sum_{i=1}^k\max_{e\in I_i}w_i(e)$…

求解带拟阵约束的最大流行匹配问题

Solving the Maximum Popular Matching Problem with Matroid Constraints

本文研究带拟阵约束的多对多匹配中最大流行匹配问题，利用新拟阵交换性质证明任意拟阵下可解，并揭示字典序偏好下存在性与验证问题为coNP-难。

Local Search for Max-Sum Diversification

We provide simple and fast polynomial time approximation schemes (PTASs) for several variants of the max-sum diversification problem which, in its most basic form, is as follows: Given n points p_1,...,p_n in R^d and an integer k, select k points such that the average Euclidean distance between these points is maximized. This problem commonly appears in information retrieval and web-search in order to select a diverse set of points from the input. In this context, it has recently received a lot of attention. We present new techniques to analyze natural local search algorithms. This leads to a …