We introduce the Multi-Objective Combinatorial Reconfiguration Optimization Problem (MO-CROP), and propose an Answer Set Programming (ASP) based approach for its solution. MO-CROP involves finding the Pareto-optimal sequences (or Pareto front) of adjacent feasible solutions between two given feasible solutions of a combinatorial problem, considering both cost and length. Our algorithm is compactly implemented through multi-shot ASP solving, and its implementing solver optirecon provides an effective tool for solving MO-CROP. As a concrete example of MO-CROP, we present an ASP encoding for solving the multi-objective independent set reconfiguration optimization problem. Experimental results on the benchmark set from the recent CoRe Challenge demonstrate our approach’s ability to capture diverse optimal sequences that reveal trade-offs between cost and length, a capability often lacking in traditional combinatorial reconfiguration methods.

Intersection graphs are well-studied in the area of graph algorithms. Some intersection graph classes are known to have algorithms enumerating all unlabeled graphs by reverse search. Since these algorithms output graphs one by one and the numbers of graphs in these classes are vast, they work only for a small number of vertices. Binary decision diagrams (BDDs) are compact data structures for various types of data and useful for solving optimization and enumeration problems. This study proposes enumeration algorithms for five intersection graph classes, which admit O ( n ) -bit string representations for their member graphs. Our algorithm for each class enumerates all unlabeled graphs with n vertices over BDDs representing the binary strings in time polynomial in n . Moreover, our algorithms are extended so that it enumerates those with constraints on the maximum (bi)clique size and/or the number of edges.

This paper presents a unified framework for capturing a variety of graph reconfiguration problems in terms of firstorder expressible properties and proposes a Boolean encoding for formulas in the first-order logic of graphs based on the exploitation of fundamental properties of graphs. We show that a variety of graph reconfiguration problems captured in our framework can be computed in a unified way by combining our encoding and Boolean satisfiability solver in a bounded model checking approach but allowing us to use quantifiers and predicates on vertices to express reconfiguration properties.

Subgraph enumeration is a fundamental task in computer science. Since the number of subgraphs can be large, some enumeration algorithms exploit compressed representations for efficiency. One such representation is the Zero-suppressed Binary Decision Diagram (ZDD). ZDDs can represent the set of subgraphs compactly and support several poly-time queries, such as counting and random sampling. Researchers have proposed efficient algorithms to construct ZDDs representing the set of subgraphs under several constraints, which yield fruitful results in many applications. Recently, Zero-suppressed Sentential Decision Diagrams (ZSDDs) have been proposed as variants of ZDDs. ZSDDs can be smaller than ZDDs when representing the same set of subgraphs. However, efficient algorithms to construct ZSDDs are known only for specific types of subgraphs: matchings and paths. We propose a novel framework to construct ZSDDs representing sets of subgraphs under given constraints. Using our framework, we can construct ZSDDs representing several sets of subgraphs such as matchings, paths, cycles, and spanning trees. We show the bound of sizes of constructed ZSDDs by the branch-width of the input graph, which is smaller than that of ZDDs by the path-width. Experiments show that our methods can construct ZSDDs faster than ZDDs and that the constructed ZSDDs are smaller than ZDDs when representing the same set of subgraphs.

It is a popular and classical problem to detect a hotspot cluster from a statistical data which is partitioned by geographical regions such as prefectures or cities. Spatial scan statistic is a standard measure of likelihood ratio which has been widely used for testing hotspot clusters. In this work, we propose a very fast algorithm to enumerate all combinatorial regions which are more significant than a given threshold value. Our algorithm features the fast exploration by pruning the search space based on the partial monotonicity of the spatial scan statistic. Experimental results for a nation-wide 47 prefectures dataset show that our method generates the highest-ranked hotspot cluster in a time a million or more times faster than the previous naive search method. Our method works practically for a dataset with several hundreds of regions, and it will drastically accelerate hotspot analysis in various fields.

Partitioning a graph into balanced components is important for several applications. For multi-objective problems, it is useful not only to find one solution but also to enumerate all the solutions with good values of objectives. However, there are a vast number of graph partitions in a graph, and thus it is difficult to enumerate desired graph partitions efficiently. In this paper, an algorithm to enumerate all the graph partitions such that all the weights of the connected components are at least a specified value is proposed. To deal with a large search space, we use zero-suppressed binary decision diagrams (ZDDs) to represent sets of graph partitions and we design a new algorithm based on frontier-based search, which is a framework to directly construct a ZDD. Our algorithm utilizes not only ZDDs but also ternary decision diagrams (TDDs) and realizes an operation which seems difficult to be designed only by ZDDs. Experimental results show that the proposed algorithm runs up to tens of times faster than an existing state-of-the-art algorithm.

Several graph libraries have been developed in the past few decades, and they were basically designed to work with a few graphs. However, there are many problems in which we have to consider all subgraphs satisfying certain constraints on a given graph. Since the number of subgraphs can increase exponentially with the graph size, explicitly representing these sets is infeasible. Hence, libraries concerned with efficiently representing a single graph instance are not suitable for such problems. In this paper, we develop Graphillion, a software library for very large sets of (vertex-)labeled graphs, based on zero-suppressed binary decision diagrams. Graphillion is not based on a traditional representation of graphs. Instead, a graph set is simply regarded as a “set of edge sets” ignoring vertices, which allows us to employ powerful tools of a “family of sets” (a set of sets) and permits large graph sets to be handled efficiently. We also utilize advanced graph enumeration algorithms, which enable the simple family tools to understand the graph structure. Graphillion is implemented as a Python library to encourage easy development of its applications, without introducing significant performance overheads. In experiments, we consider two case studies, a puzzle solver and a power network optimizer, in which several operations and heavy optimization have to be performed over very large sets of constrained graphs (i.e., cycles or forests with complicated conditions). The results show that Graphillion allows us to manage a huge number of graphs with very low development effort.

Like selective visual attention, the representation of multiple objects as an ensemble is a major strategy used by the visual system to overcome a bottleneck that the system cannot handle multiple objects simultaneously. Although simple image properties including mean size, orientation, and brightness as well as higher-level properties such as the average emotion and gender of faces, can be extracted from a group of items/images (see Alvarez, 2011 for review), it is not clear whether subjective values (e.g., attractiveness) are processed in a similar manner.