Yasushi Kawase, Atsushi Iwasaki
We consider the matching with contracts framework of Hatfield and Milgrom when one side (a firm or hospital) can make monetary transfers (offer wages) to the other (a worker or doctor). In a standard model, monetary transfers are not restricted. However, we assume that each hospital has a fixed budget; that is, the total amount of wages allocated by each hospital to the doctors is constrained. With this constraint, stable matchings may fail to exist and checking for the existence is hard. To deal with the nonexistence, we focus on near-feasible matchings that can exceed each hospital budget by a certain amount, and We introduce a new concept of compatibility. We show that the compatibility condition is a sufficient condition for the existence of a near-feasible stable matching in the matching with contracts framework. Under a slight restriction on hospitals' preferences, we provide mechanisms that efficiently return a near-feasible stable matching with respect to the actual amount of wages allocated by each hospital. By sacrificing strategy-proofness, the best possible bound of budget excess is achieved.
Keisuke Bando, Kenzo Imamura, Yasushi Kawase
Choice correspondences are crucial in decision-making, especially when faced with indifferences or ties. While tie-breaking can transform a choice correspondence into a choice function, it often introduces inefficiencies. This paper introduces a novel notion of path-independence (PI) for choice correspondences, extending the existing concept of PI for choice functions. Intuitively, a choice correspondence is PI if any consistent tie-breaking produces a PI choice function. This new notion yields several important properties. First, PI choice correspondences are rationalizabile, meaning they can be represented as the maximization of a utility function. This extends a core feature of PI in choice functions. Second, we demonstrate that the set of choices selected by a PI choice correspondence for any subset forms a generalized matroid. This property reveals that PI choice correspondences exhibit a nice structural property. Third, we establish that choice correspondences rationalized by ordinally concave functions inherently satisfy the PI condition. This aligns with recent findings that a choice function satisfies PI if and only if it can be rationalized by an ordinally concave function. Building on these theoretical foundations, we explore stable and efficient matchings under PI choice correspondences. Specifically, we investigate constrained efficient matchings, which are efficient (for one side of the market) within the set of stable matchings. Under responsive choice correspondences, such matchings are characterized by cycles. However, this cycle-based characterization fails in more general settings. We demonstrate that when the choice correspondence of each school satisfies both PI and monotonicity conditions, a similar cycle-based characterization is restored. These findings provide new insights into the matching theory and its practical applications.
Xin Han, Yasushi Kawase, Kazuhisa Makino, Haruki Yokomaku
In this paper, we introduce online knapsack problems with a resource buffer. In the problems, we are given a knapsack with capacity $1$, a buffer with capacity $R\ge 1$, and items that arrive one by one. Each arriving item has to be taken into the buffer or discarded on its arrival irrevocably. When every item has arrived, we transfer a subset of items in the current buffer into the knapsack. Our goal is to maximize the total value of the items in the knapsack. We consider four variants depending on whether items in the buffer are removable (i.e., we can remove items in the buffer) or non-removable, and proportional (i.e., the value of each item is proportional to its size) or general. For the general&non-removable case, we observe that no constant competitive algorithm exists for any $R\ge 1$. For the proportional&non-removable case, we show that a simple greedy algorithm is optimal for every $R\ge 1$. For the general&removable and the proportional&removable cases, we present optimal algorithms for small $R$ and give asymptotically nearly optimal algorithms for general $R$.
Yasushi Kawase, Atsushi Iwasaki
This paper focuses on two-sided matching where one side (a hospital or firm) is matched to the other side (a doctor or worker) so as to maximize a cardinal objective under general feasibility constraints. In a standard model, even though multiple doctors can be matched to a single hospital, a hospital has a responsive preference and a maximum quota. However, in practical applications, a hospital has some complicated cardinal preference and constraints. With such preferences (e.g., submodular) and constraints (e.g., knapsack or matroid intersection), stable matchings may fail to exist. This paper first determines the complexity of checking and computing stable matchings based on preference class and constraint class. Second, we establish a framework to analyze this problem on packing problems and the framework enables us to access the wealth of online packing algorithms so that we construct approximately stable algorithms as a variant of generalized deferred acceptance algorithm. We further provide some inapproximability results.
Yasushi Kawase
In this paper, we study twelve stochastic input models for online problems and reveal the relationships among the competitive ratios for the models. The competitive ratio is defined as the worst ratio between the expected optimal value and the expected profit of the solution obtained by the online algorithm where the input distribution is restricted according to the model. To handle a broad class of online problems, we use a framework called request-answer games that is introduced by Ben-David et al. The stochastic input models consist of two types: known distribution and unknown distribution. For each type, we consider six classes of distributions: dependent distributions, deterministic input, independent distributions, identical independent distribution, random order of a deterministic input, and random order of independent distributions. As an application of the models, we consider two basic online problems, which are variants of the secretary problem and the prophet inequality problem, under the twelve stochastic input models. We see the difference of the competitive ratios through these problems.
Yasushi Kawase, Kazuhisa Makino, Vinh Long Phan, Hanna Sumita
In this study, we investigate a scheduling problem on identical machines in which jobs require initial setup before execution. We assume that an algorithm can dynamically form a batch (i.e., a collection of jobs to be processed together) from the remaining jobs. The setup time is modeled as a known monotone function of the set of jobs within a batch, while the execution time of each job remains unknown until completion. This uncertainty poses significant challenges for minimizing the makespan. We address these challenges by considering two scenarios: each job batch must be assigned to a single machine, or a batch may be distributed across multiple machines. For both scenarios, we analyze settings with and without preemption. Across these four settings, we design online algorithms that achieve asymptotically optimal competitive ratios with respect to both the number of jobs and the number of machines.
Yasushi Kawase, Kazuhisa Makino, Hanna Sumita, Akihisa Tamura, Makoto Yokoo
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.
Yasushi Kawase, Yuko Kuroki, Atsushi Miyauchi
Aggregating responses from crowd workers is a fundamental task in the process of crowdsourcing. In cases where a few experts are overwhelmed by a large number of non-experts, most answer aggregation algorithms such as the majority voting fail to identify the correct answers. Therefore, it is crucial to extract reliable experts from the crowd workers. In this study, we introduce the notion of "expert core", which is a set of workers that is very unlikely to contain a non-expert. We design a graph-mining-based efficient algorithm that exactly computes the expert core. To answer the aggregation task, we propose two types of algorithms. The first one incorporates the expert core into existing answer aggregation algorithms such as the majority voting, whereas the second one utilizes information provided by the expert core extraction algorithm pertaining to the reliability of workers. We then give a theoretical justification for the first type of algorithm. Computational experiments using synthetic and real-world datasets demonstrate that our proposed answer aggregation algorithms outperform state-of-the-art algorithms.
Yasushi Kawase, Hanna Sumita
We study an online version of the max-min fair allocation problem for indivisible items. In this problem, items arrive one by one, and each item must be allocated irrevocably on arrival to one of $n$ agents, who have additive valuations for the items. Our goal is to make the least happy agent as happy as possible. In research on the topic of online allocation, this is a fundamental and natural problem. Our main result is to reveal the asymptotic competitive ratios of the problem for both the adversarial and i.i.d. input models. We design a polynomial-time deterministic algorithm that is asymptotically $1/n$-competitive for the adversarial model, and we show that this guarantee is optimal. To this end, we present a randomized algorithm with the same competitive ratio first and then derandomize it. A natural derandomization fails to achieve the competitive ratio of $1/n$. We instead build the algorithm by introducing a novel technique. When the items are drawn from an unknown identical and independent distribution, we construct a simple polynomial-time deterministic algorithm that outputs a nearly optimal allocation. We analyze the strict competitive ratio and show almost tight bounds for the solution. We further mention some implications of our results on variants of the problem.
Yasushi Kawase, Kazuhisa Makino, Kento Seimi
In this paper, we introduce maximum composition ordering problems. The input is $n$ real functions $f_1,\dots,f_n:\mathbb{R}\to\mathbb{R}$ and a constant $c\in\mathbb{R}$. We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation $σ:[n]\to[n]$ which maximizes $f_{σ(n)}\circ f_{σ(n-1)}\circ\dots\circ f_{σ(1)}(c)$, where $[n]=\{1,\dots,n\}$. The maximum partial composition ordering problem is to compute a permutation $σ:[n]\to[n]$ and a nonnegative integer $k~(0\le k\le n)$ which maximize $f_{σ(k)}\circ f_{σ(k-1)}\circ\dots\circ f_{σ(1)}(c)$. We propose $O(n\log n)$ time algorithms for the maximum total and partial composition ordering problems for monotone linear functions $f_i$, which generalize linear deterioration and shortening models for the time-dependent scheduling problem. We also show that the maximum partial composition ordering problem can be solved in polynomial time if $f_i$ is of form $\max\{a_ix+b_i,c_i\}$ for some constants $a_i\,(\ge 0)$, $b_i$ and $c_i$. We finally prove that there exists no constant-factor approximation algorithm for the problems, even if $f_i$'s are monotone, piecewise linear functions with at most two pieces, unless P=NP.
Yasushi Kawase, Hanna Sumita, Yu Yokoi
We investigate the problem of random assignment of indivisible goods, in which each agent has an ordinal preference and a constraint. Our goal is to characterize the conditions under which there always exists a random assignment that simultaneously satisfies efficiency and envy-freeness. The probabilistic serial mechanism ensures the existence of such an assignment for the unconstrained setting. In this paper, we consider a more general setting in which each agent can consume a set of items only if the set satisfies her feasibility constraint. Such constraints must be taken into account in student course placements, employee shift assignments, and so on. We demonstrate that an efficient and envy-free assignment may not exist even for the simple case of partition matroid constraints, where the items are categorized, and each agent demands one item from each category. We then identify special cases in which an efficient and envy-free assignment always exists. For these cases, the probabilistic serial cannot be naturally extended; therefore, we provide mechanisms to find the desired assignment using various approaches.
Toru Yoshinaga, Yasushi Kawase
Online contention resolution schemes (OCRSs) are effective rounding techniques for online stochastic combinatorial optimization problems. These schemes randomly and sequentially round a fractional solution to a relaxed problem that can be formulated in advance. In this study, we propose OCRSs for online stochastic generalized assignment problems. In the problem of our OCRSs, sequentially arriving items are packed into a single knapsack, and their sizes are revealed only after insertion. The goal of the problem is to maximize the acceptance probability, which is the smallest probability among the items being placed in the knapsack. Since the item sizes are unknown beforehand, a capacity overflow may occur. We consider two distinct settings: the hard constraint, where items that cause overflow are rejected, and the soft constraint setting, where such items are accepted. Under the hard constraint setting, we present an algorithm with an acceptance probability of $1/3$ and prove that no algorithm can achieve an acceptance probability greater than $3/7$. Under the soft constraint setting, we propose an algorithm with an acceptance probability of $1/2$ and demonstrate that this is best possible.
Yasushi Kawase, Yusuke Kobayashi, Yutaro Yamaguchi
The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where "non-zero" means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an $s$--$t$ path with two labels forbidden in a group-labeled graph. This also leads to an elementary solution to finding a zero $s$--$t$ path in a ${\mathbb Z}_3$-labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of $s$--$t$ paths in a group-labeled graph or not, and finding $s$--$t$ paths attaining at least three distinct labels if exist. The algorithm is based on a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of $s$--$t$ paths, which is the main technical contribution of this paper.
Takanori Maehara, Yasushi Kawase, Hanna Sumita, Katsuya Tono, Ken-ichi Kawarabayashi
The optimal pricing problem is a fundamental problem that arises in combinatorial auctions. Suppose that there is one seller who has indivisible items and multiple buyers who want to purchase a combination of the items. The seller wants to sell his items for the highest possible prices, and each buyer wants to maximize his utility (i.e., valuation minus payment) as long as his payment does not exceed his budget. The optimal pricing problem seeks a price of each item and an assignment of items to buyers such that every buyer achieves the maximum utility under the prices. The goal of the problem is to maximize the total payment from buyers. In this paper, we consider the case that the valuations are submodular. We show that the problem is computationally hard even if there exists only one buyer. Then we propose approximation algorithms for the unlimited budget case. We also extend the algorithm for the limited budget case when there exists one buyer and multiple buyers collaborate with each other.
Yasushi Kawase, Yutaro Yamaguchi
We explore novel connections between antimatroids and matchings in bipartite graphs. In particular, we prove that a combinatorial structure induced by stable matchings or maximum-weight matchings is an antimatroid. Moreover, we demonstrate that every antimatroid admits such a representation by stable matchings and maximum-weight matchings.
Ayumi Igarashi, Naoyuki Kamiyama, Yasushi Kawase, Warut Suksompong, Hanna Sumita, Yu Yokoi
We consider a classic many-to-one matching setting, where participants need to be assigned to teams based on the preferences of both sides. Unlike most of the matching literature, we aim to provide fairness not only to participants, but also to teams using concepts from the literature of fair division. We present a polynomial-time algorithm that computes an allocation satisfying team-justified envy-freeness up to one participant, participant-justified envy-freeness, balancedness, Pareto optimality, and group-strategyproofness for participants, even in the possible presence of ties. Our algorithm generalizes both the Gale-Shapley algorithm from two-sided matching as well as the round-robin algorithm from fair division. We also discuss how our algorithm can be extended to accommodate quotas and incomplete preferences.
Toru Yoshinaga, Yasushi Kawase
The field of algorithms with predictions aims to improve algorithm performance by integrating machine learning predictions into algorithm design. A central question in this area is how predictions can improve performance, and a key aspect of this analysis is the role of prediction accuracy. In this context, prediction accuracy is defined as a guaranteed probability that an instance drawn from the distribution belongs to the predicted set. As a performance measure that incorporates prediction accuracy, we focus on the distributionally-robust competitive ratio (DRCR), introduced by Sun et al.~(ICML 2024). The DRCR is defined as the expected ratio between the algorithm's cost and the optimal cost, where the expectation is taken over the worst-case instance distribution that satisfies the given prediction and accuracy requirement. A known structural property is that, for any fixed algorithm, the DRCR decreases linearly as prediction accuracy increases. Building on this result, we establish that the optimal DRCR value (i.e., the infimum over all algorithms) is a monotone and concave function of prediction accuracy. We further generalize the DRCR framework to a multiple-prediction setting and show that monotonicity and concavity are preserved in this setting. Finally, we apply our results to the ski rental problem, a benchmark problem in online optimization, to identify the conditions on prediction accuracies required for the optimal DRCR to attain a target value. Moreover, we provide a method for computing the critical accuracy, defined as the minimum accuracy required for the optimal DRCR to strictly improve upon the performance attainable without any accuracy guarantee.
Hiromichi Goko, Ayumi Igarashi, Yasushi Kawase, Kazuhisa Makino, Hanna Sumita, Akihisa Tamura, Yu Yokoi, Makoto Yokoo
The notion of \emph{envy-freeness} is a natural and intuitive fairness requirement in resource allocation. With indivisible goods, such fair allocations are unfortunately not guaranteed to exist. Classical works have avoided this issue by introducing an additional divisible resource, i.e., money, to subsidize envious agents. In this paper, we aim to design a truthful allocation mechanism of indivisible goods to achieve both fairness and efficiency criteria with a limited amount of subsidy. Following the work of Halpern and Shah, our central question is as follows: to what extent do we need to rely on the power of money to accomplish these objectives? For general valuations, the impossibility theorem of combinatorial auction translates to our setting: even if an arbitrarily large amount of money is available for use, no mechanism can achieve truthfulness, envy-freeness, and utilitarian optimality simultaneously when agents have general monotone submodular valuations. By contrast, we show that, when agents have matroidal valuations, there is a truthful allocation mechanism that achieves envy-freeness and utilitarian optimality by subsidizing each agent with at most $1$, the maximum marginal contribution of each item for each agent. The design of the mechanism rests crucially on the underlying matroidal M-convexity of the Lorenz dominating allocations.
Yuval Filmus, Yasushi Kawase, Yusuke Kobayashi, Yutaro Yamaguchi
A set function is called XOS if it can be represented by the maximum of additive functions. When such a representation is fixed, the number of additive functions required to define the XOS function is called the width. In this paper, we study the problem of maximizing XOS functions in the value oracle model. The problem is trivial for the XOS functions of width $1$ because they are just additive, but it is already nontrivial even when the width is restricted to $2$. We show two types of tight bounds on the polynomial-time approximability for this problem. First, in general, the approximation bound is between $O(n)$ and $Ω(n / \log n)$, and exactly $Θ(n / \log n)$ if randomization is allowed, where $n$ is the ground set size. Second, when the width of the input XOS functions is bounded by a constant $k \geq 2$, the approximation bound is between $k - 1$ and $k - 1 - ε$ for any $ε> 0$. In particular, we give a linear-time algorithm to find an exact maximizer of a given XOS function of width $2$, while we show that any exact algorithm requires an exponential number of value oracle calls even when the width is restricted to $3$.
Yasushi Kawase, Koichi Nishimura, Hanna Sumita
The fair allocation of mixed goods, consisting of both divisible and indivisible goods, has been a prominent topic of study in economics and computer science. We define an allocation as fair if its utility vector minimizes a symmetric strictly convex function. This fairness criterion includes standard ones such as maximum egalitarian social welfare and maximum Nash social welfare. We address the problem of minimizing a given symmetric strictly convex function when agents have binary valuations. If only divisible goods or only indivisible goods exist, the problem is known to be solvable in polynomial time. In this paper, firstly, we demonstrate that the problem is NP-hard even when all indivisible goods are identical. This NP-hardness is established even for maximizing egalitarian social welfare or Nash social welfare. Secondly, we provide a polynomial-time algorithm for the problem when all divisible goods are identical. To accomplish these, we exploit the proximity structure inherent in the problem. This provides theoretically important insights into the hybrid domain of convex optimization that incorporates both discrete and continuous aspects.