A Fast Maximum $k$-Plex Algorithm Parameterized by the Degeneracy Gap

Given a graph, the $k$-plex is a vertex set in which each vertex is not adjacent to at most $k-1$ other vertices in the set. The maximum $k$-plex problem, which asks for the largest $k$-plex from a given graph, is an important but computationally challenging problem in applications like graph search and community detection. So far, there is a number of empirical algorithms without sufficient theoretical explanations on the efficiency. We try to bridge this gap by defining a novel parameter of the input instance, $g_k(G)$, the gap between the degeneracy bound and the size of maximum $k$-plex in the given graph, and presenting an exact algorithm parameterized by $g_k(G)$. In other words, we design an algorithm with running time polynomial in the size of input graph and exponential in $g_k(G)$ where $k$ is a constant. Usually, $g_k(G)$ is small and bounded by $O(\log{(|V|)})$ in real-world graphs, indicating that the algorithm runs in polynomial time. We also carry out massive experiments and show that the algorithm is competitive with the state-of-the-art solvers. Additionally, for large $k$ values such as $15$ and $20$, our algorithm has superior performance over existing algorithms.

Facility Location with Entrance Fees

In mechanism design, the facility location game is an extensively studied problem. In the classical model, the cost of each agent is her distance to the nearest facility. In this paper, we consider a new model, where there is a location-dependent entrance fee to the facility. Thus, in our model, the cost of each agent is the sum of the distance to the facility and the entrance fee of the facility. This is a refined generalization of the classical model. We study the model and design strategyproof mechanisms. For one and two facilities, we provide upper and lower bounds for the approximation ratio given by deterministic and randomized mechanisms, with respect to the utilitarian objective and the egalitarian objective. Most of our bounds are tight and these bounds are independent of the entrance fee functions. Our results are as general as possible because the entrance fee function we consider is arbitrary.

Listing Maximal k-Plexes in Large Real-World Graphs

Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal $k$-plexes in $O^*(\gamma^D)$ time for each constant $k$, where $\gamma$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less than the vertex number $n$ in real-word graphs. Compared to the trivial bound of $2^n$, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing $k$-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.

Enhancing Balanced Graph Edge Partition with Effective Local Search

Graph partition is a key component to achieve workload balance and reduce job completion time in parallel graph processing systems. Among the various partition strategies, edge partition has demonstrated more promising performance in power-law graphs than vertex partition and thereby has been more widely adopted as the default partition strategy by existing graph systems. The graph edge partition problem, which is to split the edge set into multiple balanced parts to minimize the total number of copied vertices, has been widely studied from the view of optimization and algorithms. In this paper, we study local search algorithms for this problem to further improve the partition results from existing methods. More specifically, we propose two novel concepts, namely adjustable edges and blocks. Based on these, we develop a greedy heuristic as well as an improved search algorithm utilizing the property of the max-flow model. To evaluate the performance of our algorithms, we first provide adequate theoretical analysis in terms of the approximation quality. We significantly improve the previously known approximation ratio for this problem. Then we conduct extensive experiments on a large number of benchmark datasets and state-of-the-art edge partition strategies. The results show that our proposed local search framework can further improve the quality of graph partition by a wide margin.

Object Reachability via Swaps under Strict and Weak Preferences

The \textsc{Housing Market} problem is a widely studied resource allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We first consider whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie among objects is allowed). This problem is polynomially solvable in a star-network and NP-complete in a tree-network. It is left as a challenging open problem whether the problem is polynomially solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Then we show that when the preferences of the agents are weak (ties among objects are allowed), the problem becomes NP-hard even when the network is a path. In addition, we consider the computational complexity of finding different optimal assignments for the problem under the network being a path or a star.