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The flow game with public arcs is a cooperative revenue game derived from a flow network. In this game, each player possesses an arc, while certain arcs, known as public arcs, are not owned by any specific player and are accessible to any coalition. The aim of this game is to maximize the flow that can be routed in the network through strategic coalition formation. By exploring its connection to the maximum partially disjoint path problem, we investigate the approximate core and nucleon of the flow game with public arcs. The approximate core is an extension of the core that allows for some deviation in group rationality, while the nucleon is a multiplicative analogue of the nucleolus. In this paper, we provide two complete characterizations for the optimal approximate core and show that the nucleon can be computed in polynomial time.

We study the approximate core for edge cover games, which are cooperative games stemming from edge cover problems. In these games, each player controls a vertex on a network $G = (V, E; w)$, and the cost of a coalition $S\subseteq V$ is equivalent to the minimum weight of edge covers in the subgraph induced by $S$. We prove that the 3/4-core of edge cover games is always non-empty and can be computed in polynomial time by using linear program duality approach. This ratio is the best possible, as it represents the integrality gap of the natural LP for edge cover problems. Moreover, our analysis reveals that the ratio of approximate core corresponds with the length of the shortest odd cycle of underlying graphs.

In this paper, we propose a constrained heterogeneous facility location model where a set of alternative locations are feasible for building facilities and the number of facilities built at each location is limited. Supposing that a set of agents on the real line can strategically report their locations and each agent's cost is her distance to the further facility that she is interested in, we study deterministic mechanism design without money for constrained heterogeneous two-facility location games. Depending on whether agents have optional preference, the problem is considered in two settings: the compulsory setting and the optional setting. In the compulsory setting where each agent is served by the two heterogeneous facilities, we provide a 3-approximate deterministic group strategyproof mechanism for the sum/maximum cost objective respectively, which is also the best deterministic strategyproof mechanism under the corresponding social objective. In the optional setting where each agent can be interested in one of the two facilities or both, we propose a deterministic group strategyproof mechanism with approximation ratio of at most $2n+1$ for the sum cost objective and a deterministic group strategyproof mechanism with approximation ratio of at most 9 for the maximum cost objective., Comment: 19 pages

The matching game is a cooperative game where the value of every coalition is the maximum revenue of players in the coalition can make by forming pairwise disjoint partners. The multiple partners matching game generalizes the matching game by allowing each player to have more than one possibly repeated partner. In this paper, we study profit-sharing in multiple partners matching games. A central concept for profit-sharing is the core which consists of all possible ways of distributing the profit among individual players such that the grand coalition remains intact. The core of multiple partners matching games may be empty [Deng et al., Algorithmic aspects of the core of combinatorial optimization games, Math. Oper. Res., 1999.]; even when the core is non-empty, the core membership problem is intractable in general [Biro et al., The stable fixtures problem with payments, Games Econ. Behav., 2018]. Thus we study approximate core allocations upon which a coalition may be paid less than the profit it makes by seceding from the grand coalition. We provide an LP-based mechanism guaranteeing that no coalition is paid less than $2/3$ times the profit it makes on its own. We also show that $2/3$ is the best possible factor relative to the underlying LP-relaxation. Our result generalizes the work of Vazirani [Vazirani, The general graph matching game: approximate core, arXiv, 2021] from matching games to multiple partners matching games., Comment: 12 pages

A matching game is a cooperative profit game defined on an edge-weighted graph, where the players are the vertices and the profit of a coalition is the maximum weight of matchings in the subgraph induced by the coalition. A population monotonic allocation scheme is a collection of rules defining how to share the profit among players in each coalition such that every player is better off when the coalition expands. In this paper, we study matching games and provide a necessary and sufficient characterization for the existence of population monotonic allocation schemes. Our characterization also implies that whether a matching game admits population monotonic allocation schemes can be determined efficiently.

Online social networks have been one of the most effective platforms for marketing and advertising. Through "word of mouth" effects, information or product adoption could spread from some influential individuals to millions of users in social networks. Given a social network $G$ and a constant $k$, the influence maximization problem seeks for $k$ nodes in $G$ that can influence the largest number of nodes. This problem has found important applications, and a large amount of works have been devoted to identifying the few most influential users. But most of existing works only focus on the diffusion of a single idea or product in social networks. However, in reality, one company may produce multiple kinds of products and one user may also have multiple adoptions. Given multiple kinds of different products with different activation costs and profits, it is crucial for the company to distribute the limited budget among multiple products in order to achieve profit maximization. Profit Maximization with Multiple Adoptions (PM$^{2}$A) problem aims to seek for a seed set within the budget to maximize the overall profit. In this paper, a Randomized Modified Greedy (RMG) algorithm based on the Reverse Influence Sampling (RIS) technique is presented for the PM$^{2}$A problem, which could achieve a $(1-1/e-\varepsilon)$-approximate solution with high probability. Compared with the algorithm proposed in [16] that achieves a $\frac{1}{2}(1-1/e^{2})$-approximate solution, our algorithm provides a better performance ratio which is also the best performance ratio of the PM$^{2}$A problem. Comprehensive experiments on three real-world social networks are conducted, and the results demonstrate that our RMG algorithm outperforms the algorithm proposed in [16] and other heuristics in terms of profit maximization, and could better allocate the budget., Comment: 21 pages, 6 figures

The arboricity of a graph is the minimum number of forests required to cover all its edges. In this paper, we examine arboricity from a game-theoretic perspective and investigate cost-sharing in the minimum forest cover problem. We introduce the arboricity game as a cooperative cost game defined on a graph. The players are edges, and the cost of each coalition is the arboricity of the subgraph induced by the coalition. We study properties of the core and propose an efficient algorithm for computing the nucleolus when the core is not empty. In order to compute the nucleolus in the core, we introduce the prime partition which is built on the densest subgraph lattice. The prime partition decomposes the edge set of a graph into a partially ordered set defined from minimal densest minors and their invariant precedence relation. Moreover, edges from the same partition always have the same value in a core allocation. Consequently, when the core is not empty, the prime partition significantly reduces the number of variables and constraints required in the linear programs of Maschler's scheme and allows us to compute the nucleolus in polynomial time. Besides, the prime partition provides a graph decomposition analogous to the celebrated core decomposition and the density-friendly decomposition, which may be of independent interest.

For vertex cover games (introduced by Deng et al., Math. Oper. Res., 24:751-766, 1999), we investigate population monotonic allocation schemes (introduced by Sprumont, Games Econ. Behav., 2: 378-394, 1990). We show that the existence of a population monotonic allocation scheme (PMAS for short) for vertex cover games can be determined efficiently and that a PMAS, if exists, can be constructed accordingly. We also show that integral PMAS-es for vertex cover games can be characterized with stable matchings and be enumerated by employing Gale-Shapley algorithm (introduced by Gale and Shapley, Amer. Math. Monthly, 69:9-15, 1962)., Comment: To appear in Theoretical Computer Science

Independent set games are cooperative games defined on graphs, where players are edges and the value of a coalition is the maximum cardinality of independent sets in the subgraph defined by the coalition. In this paper, we investigate the convexity of independent set games, as convex games possess many nice properties both economically and computationally. For independent set games introduced by Deng et al. (Math. Oper. Res., 24:751-766, 1999), we provide a necessary and sufficient characterization for the convexity, i.e., every non-pendant edge is incident to a pendant edge in the underlying graph. Our characterization immediately yields a polynomial time algorithm for recognizing convex instances of independent set games. Besides, we introduce a new class of independent set games and provide an efficient characterization for the convexity.

Cooperative games provide an appropriate framework for fair and stable profit distribution in multiagent systems. In this paper, we study the algorithmic issues on path cooperative games that arise from the situations where some commodity flows through a network. In these games, a coalition of edges or vertices is successful if it enables a path from the source to the sink in the network, and lose otherwise. Based on dual theory of linear programming and the relationship with flow games, we provide the characterizations on the CS-core, least-core and nucleolus of path cooperative games. Furthermore, we show that the least-core and nucleolus are polynomially solvable for path cooperative games defined on both directed and undirected network.

Cooperative games provide a framework for fair and stable profit allocation in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are such solution concepts that characterize stability of cooperation. In this paper, we study the algorithmic issues on the least-core and nucleolus of threshold cardinality matching games (TCMG). A TCMG is defined on a graph $G=(V,E)$ and a threshold $T$, in which the player set is $V$ and the profit of a coalition $S\subseteq V$ is 1 if the size of a maximum matching in $G[S]$ meets or exceeds $T$, and 0 otherwise. We first show that for a TCMG, the problems of computing least-core value, finding and verifying least-core payoff are all polynomial time solvable. We also provide a general characterization of the least core for a large class of TCMG. Next, based on Gallai-Edmonds Decomposition in matching theory, we give a concise formulation of the nucleolus for a typical case of TCMG which the threshold $T$ equals $1$. When the threshold $T$ is relevant to the input size, we prove that the nucleolus can be obtained in polynomial time in bipartite graphs and graphs with a perfect matching.