Randomized Strategies for Robust Combinatorial Optimization

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. Our goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update method. A key technique is to introduce a new concept called $(\eta,\gamma)$-reductions for objective functions with parameters $\eta, \gamma$. We show that our scheme outputs a nearly $\alpha$-approximate solution if there exists an $\alpha$-approximation algorithm for a subproblem defined by $(\eta,\gamma)$-reductions. This improves approximation ratio in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem

Envy-freeness and maximum Nash welfare for mixed divisible and indivisible goods

We study fair allocation of resources consisting of both divisible and indivisible goods to agents with additive valuations. When only divisible or indivisible goods exist, it is known that an allocation that achieves the maximum Nash welfare (MNW) satisfies the classic fairness notions based on envy. In addition, properties of the MNW allocations for binary valuations are known. In this paper, we show that when all agents' valuations are binary and linear for each good, an MNW allocation for mixed goods satisfies the envy-freeness up to any good for mixed goods. This notion is stronger than an existing one called envy-freeness for mixed goods (EFM), and our result generalizes the existing results for the case when only divisible or indivisible goods exist. Moreover, our result holds for a general fairness notion based on minimizing a symmetric strictly convex function. For the general additive valuations, we also provide a formal proof that an MNW allocation satisfies a weaker notion than EFM

Parameterized Complexity of Sparse Linear Complementarity Problems

In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with all the parameters. We also show that if we drop any of the three parameters, then the LCP is fixed-parameter intractable. In addition, we discuss the nonexistence of a polynomial kernel for the LCP

Stochastic Solutions for Dense Subgraph Discovery in Multilayer Networks

Network analysis has played a key role in knowledge discovery and data mining. In many real-world applications in recent years, we are interested in mining multilayer networks, where we have a number of edge sets called layers, which encode different types of connections and/or time-dependent connections over the same set of vertices. Among many network analysis techniques, dense subgraph discovery, aiming to find a dense component in a network, is an essential primitive with a variety of applications in diverse domains. In this paper, we introduce a novel optimization model for dense subgraph discovery in multilayer networks. Our model aims to find a stochastic solution, i.e., a probability distribution over the family of vertex subsets, rather than a single vertex subset, whereas it can also be used for obtaining a single vertex subset. For our model, we design an LP-based polynomial-time exact algorithm. Moreover, to handle large-scale networks, we also devise a simple, scalable preprocessing algorithm, which often reduces the size of the input networks significantly and results in a substantial speed-up. Computational experiments demonstrate the validity of our model and the effectiveness of our algorithms.Comment: Accepted to WSDM 202

Fair Allocation with Binary Valuations for Mixed Divisible and Indivisible Goods

The fair allocation of mixed goods, consisting of both divisible and indivisible goods, among agents with heterogeneous preferences, has been a prominent topic of study in economics and computer science. In this paper, we investigate the nature of fair allocations when agents have binary valuations. We define an allocation as fair if its utility vector minimizes a symmetric strictly convex function, which includes conventional fairness criteria such as maximum egalitarian social welfare and maximum Nash social welfare. While a good structure is known for the continuous case (where only divisible goods exist) or the discrete case (where only indivisible goods exist), deriving such a structure in the hybrid case remains challenging. Our contributions are twofold. First, we demonstrate that the hybrid case does not inherit some of the nice properties of continuous or discrete cases, while it does inherit the proximity theorem. Second, we analyze the computational complexity of finding a fair allocation of mixed goods based on the proximity theorem. In particular, we provide a polynomial-time algorithm for the case when all divisible goods are identical and homogeneous, and demonstrate that the problem is NP-hard in general. Our results also contribute to a deeper understanding of the hybrid convex analysis

Towards Optimal Subsidy Bounds for Envy-freeable Allocations

We study the fair division of indivisible items with subsidies among $n$ agents, where the absolute marginal valuation of each item is at most one. Under monotone valuations (where each item is a good), Brustle et al. (2020) demonstrated that a maximum subsidy of $2(n-1)$ and a total subsidy of $2(n-1)^2$ are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most $n-1$ per agent and a total subsidy of at most $n(n-1)/2$. Moreover, we present further improved bounds for monotone valuations.Comment: 14page

線形相補性問題 : 計算複雑度と整数性

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 室田 一雄, 東京大学教授 岩田 覚, 東京大学准教授 平井 広志, 京都大学准教授 牧野 和久, 日本大学准教授 森山 園子University of Tokyo(東京大学

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}^kmax_{ein I_i}w_i(e)), we show that the problem is strongly NP-hard but admits a PTAS