Hedonic Games and Treewidth Revisited

We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph G = (V,E), and the weight of an arc uv denotes the utility u gains by being in the same coalition as v. We focus on (arguably) the most basic stability question about such a game: given a graph, does a Nash stable solution exist and can we find it efficiently? We study the (parameterized) complexity of ASHG stability when the underlying graph has treewidth t and maximum degree ?. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly 2^{O(??t)}. We present an algorithm with parameter dependence (? t)^{O(? t)}, significantly improving upon the parameter dependence on ? given by Peters, albeit with a slightly worse dependence on t. Our main result is that this slight performance deterioration with respect to t is actually completely justified: we observe that the previously claimed algorithm is incorrect, and that in fact no algorithm can achieve dependence t^{o(t)} for bounded-degree graphs, unless the ETH fails. This, together with corresponding bounds we provide on the dependence on ? and the joint parameter establishes that our algorithm is essentially optimal for both parameters, under the ETH. We then revisit the parameterization by treewidth alone and resolve a question also posed by Peters by showing that Nash Stability remains strongly NP-hard on stars under additive preferences. Nevertheless, we also discover an island of mild tractability: we show that Connected Nash Stability is solvable in pseudo-polynomial time for constant t, though with an XP dependence on t which, as we establish, cannot be avoided

On Directed Covering and Domination Problems

In this paper, we study covering and domination problems on directed graphs. Although undirected Vertex Cover and Edge Dominating Set are well-studied classical graph problems, the directed versions have not been studied much due to the lack of clear definitions. We give natural definitions for Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set as directed generations of Vertex Cover and Edge Dominating Set. For these problems, we show that (1) Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set are NP-complete on planar directed acyclic graphs except when r=1 or (p,q)=(0,0), (2) if r>=2, Directed r-In (Out) Vertex Cover is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs, (3) if either p or q is greater than 1, Directed (p,q)-Edge Dominating Set is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs, (4) all problems can be solved in polynomial time on trees, and (5) Directed (0,1),(1,0),(1,1)-Edge Dominating Set are fixed-parameter tractable in general graphs. The first result implies that (directed) r-Dominating Set on directed line graphs is NP-complete even if r=1

A risk analysis on geographical concentration of global supply chains

In this paper we present and analyze new referential statistics for risk assessment on geographical concentration of global supply chains. The study’s net contribution rests on the development of a metric which indicates geographical concentration in terms of the frequency of supply chain engagement with the regions of analytical concerns, alongside the traditional approach based on volume measures of value-added concentration. Japan, a country with a high propensity to encounter natural hazards, and China, under a mounting geopolitical tension with the United States, are chosen as target regions for the risk assessment. The analysis follows a line of techniques in input-output economics known as the “key sector analysis”, yet with methodological augmentation by a compatible analytical framework in the network theory. Using the latest set of multi-country input-output tables constructed by the Organisation for Economic Co-operation and Development (OECD), the concentration risks of some key global supply chains such as the automotive industry and the ICT/electronics equipment industry are identified

Capacitated Network Design Games on a Generalized Fair Allocation Model

The cost-sharing connection game is a variant of routing games on a network. In this model, given a directed graph with edge-costs and edge-capacities, each agent wants to construct a path from a source to a sink with low cost. The cost of each edge is shared by the users based on a cost-sharing function. One of simple cost-sharing functions is defined as the cost divided by the number of users. In fact, most of the previous papers about cost-sharing connection games addressed this cost-sharing function. It models an ideal setting, where no overhead arises when people share things, though it might be quite rare in real life; it is more realistic to consider the setting that the cost paid by an agent is the original cost per the number of the agents plus the overhead. In this paper, we model the more realistic scenario of cost-sharing connection games by generalizing cost-sharing functions. The arguments on the model do not depend on specific cost-sharing functions, and are applicable for a wide class of cost-sharing functions satisfying the following natural properties: they are (1) non-increasing, (2) lower bounded by the original cost per the number of the agents, and (3) upper bounded by the original cost, which enables to represent various scenarios of cost-sharing. We investigate the Price of Anarchy (PoA) and the Price of Stability (PoS) under sum-cost and max-cost criteria with the generalized cost-sharing function. In spite of the generalization, we obtain the same bounds of PoA and PoS as the cost-sharing with no overhead except PoS under sum-cost. Note that these bounds are tight. In the case of sum-cost, the lower bound on PoS increases from $\log n$ to $n+1/n-1$ by the generalization, which is also almost tight because the upper bound is $n$.Comment: 13 pages, 2 figure

Shortest Beer Path Queries based on Graph Decomposition

Given a directed edge-weighted graph $G=(V, E)$ with beer vertices $B\subseteq V$, a beer path between two vertices $u$ and $v$ is a path between $u$ and $v$ that visits at least one beer vertex in $B$, and the beer distance between two vertices is the shortest length of beer paths. We consider \emph{indexing problems} on beer paths, that is, a graph is given a priori, and we construct some data structures (called indexes) for the graph. Then later, we are given two vertices, and we find the beer distance or beer path between them using the data structure. For such a scheme, efficient algorithms using indexes for the beer distance and beer path queries have been proposed for outerplanar graphs and interval graphs. For example, Bacic et al. (2021) present indexes with size $O(n)$ for outerplanar graphs and an algorithm using them that answers the beer distance between given two vertices in $O(\alpha(n))$ time, where $\alpha(\cdot)$ is the inverse Ackermann function; the performance is shown to be optimal. This paper proposes indexing data structures and algorithms for beer path queries on general graphs based on two types of graph decomposition: the tree decomposition and the triconnected component decomposition. We propose indexes with size $O(m+nr^2)$ based on the triconnected component decomposition, where $r$ is the size of the largest triconnected component. For a given query $u,v\in V$, our algorithm using the indexes can output the beer distance in query time $O(\alpha(m))$. In particular, our indexing data structures and algorithms achieve the optimal performance (the space and the query time) for series-parallel graphs, which is a wider class of outerplanar graphs.Comment: 25 pages, 9 figure

Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond

A dominating set $S$ of graph $G$ is called an $r$-grouped dominating set if $S$ can be partitioned into $S_1,S_2,\ldots,S_k$ such that the size of each unit $S_i$ is $r$ and the subgraph of $G$ induced by $S_i$ is connected. The concept of $r$-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets ($r=1$), paired dominating sets ($r=2$), and connected dominating sets ($r$ is arbitrary and $k=1$). In this paper, we investigate the computational complexity of $r$-Grouped Dominating Set, which is the problem of deciding whether a given graph has an $r$-grouped dominating set with at most $k$ units. For general $r$, the problem is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which $r$ is a constant or a parameter, but we see that the problem for every fixed $r>0$ is still hard to solve. From the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that it is fixed-parameter tractable for $r$ and treewidth, because the condition of $r$-grouped domination for a constant $r$ can be represented as monadic second-order logic (mso2). This is good news, but the running time is not practical. We then design an $O^*(\min\{(2\tau(r+1))^{\tau},(2\tau)^{2\tau}\})$-time algorithm for general $r\ge 2$, where $\tau$ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., $r \in \{2,3\}$, we can speed up the algorithm, whose running time becomes $O^*((r+1)^\tau)$. We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of $r$-Grouped Dominating Set.Comment: 23 pages, 6 figure