Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Several concepts of fairness have been established, and envy-freeness is one of the most widely studied notions of fairness. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is only known that an EFX allocation always exists when there are at most three agents and their valuations are additive or when all agents have identical valuations.In this paper, we show that an EFX allocation always exists when every agent has one of the two additive valuations. We give a constructive proof, in which we iteratively obtain a Pareto dominating EFX allocation from an existing EFX allocation.

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Envy-freeness is one of the most widely studied notions in fair division. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling concept is envy-freeness up to any item (EFX). We study the existence of EFX allocations for general valuations. The existence of EFX allocations is a major open problem. For general valuations, it is known that an EFX allocation always exists (i) when n = 2 or (ii) when all agents have identical valuations, where n is the number of agents. it is also known that an EFX allocation always exists when one can leave at most n − 1 items unallocated.We develop new techniques and extend some results of additive valuations to general valuations on the existence of EFX allocations. We show that an EFX allocation always exists (i) when all agents have one of two general valuations or (ii) when the number of items is at most n + 3. We also show that an EFX allocation always exists when one can leave at most n − 2 items unallocated. In addition to the positive results, we construct an instance with n = 3 in which an existing approach does not work as it is.

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The Steiner tree problem is one of the most fundamental combinatorial optimization problems. In this problem, the input is an undirected graph, a cost function on the edge set, and a subset of vertices called terminals, and the objective is to find a minimum-cost tree spanning all the terminals. By changing the objective function, several variants of the Steiner tree problem have been studied in the literature. For example, in the min-power version of the problem, the goal is to find a Steiner tree S minimizing the total power consumption of vertices, where the power of a vertex v is the maximum cost of any edge of S incident to v. Another example is the node-weighted version of the problem, in which costs are assigned to vertices instead of edges. In this paper, we introduce a common generalization of these variants and give an approximation algorithm for it. When the maximum degree ∆ of the input graph is constant, our algorithm runs in polynomial time and the approximation ratio is 2.016 + ln ∆.