Storing Set Families More Compactly with Top ZDDs

Zero-suppressed Binary Decision Diagrams (ZDDs) are data structures for representing set families in a compressed form. With ZDDs, many valuable operations on set families can be done in time polynomial in ZDD size. In some cases, however, the size of ZDDs for representing large set families becomes too huge to store them in the main memory. This paper proposes top ZDD, a novel representation of ZDDs which uses less space than existing ones. The top ZDD is an extension of top tree, which compresses trees, to compress directed acyclic graphs by sharing identical subgraphs. We prove that navigational operations on ZDDs can be done in time poly-logarithmic in ZDD size, and show that there exist set families for which the size of the top ZDD is exponentially smaller than that of the ZDD. We also show experimentally that our top ZDDs have smaller size than ZDDs for real data

International Competition on Graph Counting Algorithms 2023

This paper reports on the details of the International Competition on Graph Counting Algorithms (ICGCA) held in 2023. The graph counting problem is to count the subgraphs satisfying specified constraints on a given graph. The problem belongs to #P-complete, a computationally tough class. Since many essential systems in modern society, e.g., infrastructure networks, are often represented as graphs, graph counting algorithms are a key technology to efficiently scan all the subgraphs representing the feasible states of the system. In the ICGCA, contestants were asked to count the paths on a graph under a length constraint. The benchmark set included 150 challenging instances, emphasizing graphs resembling infrastructure networks. Eleven solvers were submitted and ranked by the number of benchmarks correctly solved within a time limit. The winning solver, TLDC, was designed based on three fundamental approaches: backtracking search, dynamic programming, and model counting or #SAT (a counting version of Boolean satisfiability). Detailed analyses show that each approach has its own strengths, and one approach is unlikely to dominate the others. The codes and papers of the participating solvers are available: https://afsa.jp/icgca/.Comment: https://afsa.jp/icgca

Finding the Anticover of a String

A k-anticover of a string x is a set of pairwise distinct factors of x of equal length k, such that every symbol of x is contained into an occurrence of at least one of those factors. The existence of a k-anticover can be seen as a notion of non-redundancy, which has application in computational biology, where they are associated with various non-regulatory mechanisms. In this paper we address the complexity of the problem of finding a k-anticover of a string x if it exists, showing that the decision problem is NP-complete on general strings for k ? 3. We also show that the problem admits a polynomial-time solution for k=2. For unbounded k, we provide an exact exponential algorithm to find a k-anticover of a string of length n (or determine that none exists), which runs in O*(min {3^{(n-k)/3)}, ((k(k+1))/2)^{n/(k+1)) time using polynomial space

Variable Shift SDD: A More Succinct Sentential Decision Diagram

The Sentential Decision Diagram (SDD) is a tractable representation of Boolean functions that subsumes the famous Ordered Binary Decision Diagram (OBDD) as a strict subset. SDDs are attracting much attention because they are more succinct than OBDDs, as well as having canonical forms and supporting many useful queries and transformations such as model counting and Apply operation. In this paper, we propose a more succinct variant of SDD named Variable Shift SDD (VS-SDD). The key idea is to create a unique representation for Boolean functions that are equivalent under a specific variable substitution. We show that VS-SDDs are never larger than SDDs and there are cases in which the size of a VS-SDD is exponentially smaller than that of an SDD. Moreover, despite such succinctness, we show that numerous basic operations that are supported in polytime with SDD are also supported in polytime with VS-SDD. Experiments confirm that VS-SDDs are significantly more succinct than SDDs when applied to classical planning instances, where inherent symmetry exists

系列二分決定グラフを用いた部分文字列索引の構築

ERATO 湊離散構造処理系プロジェクト春のワークショップ（キックオフシンポジウム）. 2010年5月28日（金）～29日（土）. ERATO湊プロジェクト研究室

Counterexamples to the long-standing conjecture on the complexity of BDD binary operations

In this article, we disprove the long-standing conjecture, proposed by R.E. Bryant in 1986, that his binary decision diagram (BDD) algorithm computes any binary operation on two Boolean functions in linear time in the input-output sizes. We present Boolean functions for which the time required by Bryant's algorithm is a quadratic of the input-output sizes for all nontrivial binary operations, such as ⋀, ⋁, and ⊕. For the operations ⋀ and ⋁, we show an even stronger counterexample where the output BDD size is constant, but the computation time is still a quadratic of the input BDD size. In addition, we present experimental results to support our theoretical observations