Incremental Polynomial Time Research Articles

Representations for the Largest Extension of a Closure System

An extension of a closure system on a finite set S is a closure system on the same set S containing the first one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set of its meet-irreducible elements. When a closure system is described by an implicational base, we provide a characterization of the implicational base for the largest extension. We also show that the largest extension can be handled by a small modification of the implicational base of the input closure system. This answers a question asked in Yoshikawa et al. (J. Math. Psychol. 77, 82–93, 2017). Second, we are interested in computing the largest extension when the closure system is given by the set of all its meet-irreducible elements. We give an incremental polynomial time algorithm to compute the largest extension of a closure system, and left open whether the number of meet-irreducible elements grows exponentially.

Generating clause sequences of a CNF formula

Given a CNF formula Φ with clauses C1,…,Cm and variables V={x1,…,xn}, a truth assignment a:V→{0,1} of Φ leads to a clause sequence σΦ(a)=(C1(a),…,Cm(a))∈{0,1}m where Ci(a)=1 if clause Ci evaluates to 1 under assignment a, otherwise Ci(a)=0. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties.We consider a problem posed at Dagstuhl Seminar 19211 “Enumeration in Data Management” (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.

Efficiently enumerating minimal triangulations

We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where “proper” means that the tree decomposition cannot be improved by removing or splitting a bag. The algorithm can incorporate any method for (ordinary, single result) triangulation or tree decomposition, and can serve as an anytime algorithm to improve such a method. We describe an extensive experimental study of an implementation on real data from different fields. Our experiments show that the algorithm improves upon central quality measures over the underlying tree decompositions, and is able to produce a large number of high-quality decompositions.

Enumerating Vertices of Covering Polyhedra with Totally Unimodular Constraint Matrices

We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron $P=P(A,\b1)=\{x\in \mathbb{R}^n \mid Ax\geq \b1,~x\geq \b0\}$, when $A$ is a totally unimodular matrix. Our algorithm is based on decomposing the hypergraph transversal problem for unimodular hypergraphs using Seymour's decomposition of totally unimodular matrices and may be of independent interest.

A global parallel algorithm for enumerating minimal transversals of geometric hypergraphs

We consider the problem of enumerating all minimal hitting sets of a given hypergraph (V,R), where V is a finite set, called the vertex set and R is a set of subsets of V called the hyperedges. We show that, when the hypergraph admits a balanced subdivision, then a recursive decomposition can be used to obtain efficiently all minimal hitting sets of the hypergraph. We apply this decomposition framework to get incremental polynomial-time algorithms for finding minimal hitting sets and minimal set covers for a number of hypergraphs induced by a set of points and a set of geometric objects. The set of geometric objects includes half-spaces, hyper-rectangles and balls, in fixed dimension. A distinguishing feature of the algorithms we obtain is that they admit an efficient global parallel implementation, in the sense that all minimal hitting sets can be generated in polylogarithmic time in |V|, |R| and the total number of minimal transversals M, using a polynomial number of processors. For half-spaces in R2, we show that the above two enumeration problems can be solved, using a different technique, with amortized polynomial delay.

Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width

The linear induced matching width (LMIM-width) of a graph is a width parameter defined by using the notion of branch-decompositions of a set function on ternary trees. In this paper we study output-polynomial enumeration algorithms on graphs of bounded LMIM-width and graphs of bounded local LMIM-width. In particular, we show that all 1-minimal and all 1-maximal $$(\sigma ,\rho )$$ -dominating sets, and hence all minimal dominating sets, of graphs of bounded LMIM-width can be enumerated with polynomial (linear) delay using polynomial space. Furthermore, we show that all minimal dominating sets of a unit square graph can be enumerated in incremental polynomial time.

Naive Infinite Enumeration of Context-free Languages in Incremental Polynomial Time

We consider the naive bottom-up concatenation scheme for a con- text-free language and show that this scheme has the incremental polynomial time property. This means that all members of the language can be enumerated without duplicates so that the time between two consecutive outputs is bounded by a polynomial in the number of strings already generated.

An Incremental Polynomial Time Algorithm to Enumerate All Minimal Edge Dominating Sets

We show that all minimal edge dominating sets of a graph can be generated in incremental polynomial time. We present an algorithm that solves the equivalent problem of enumerating minimal (vertex) dominating sets of line graphs in incremental polynomial, and consequently output polynomial, time. Enumeration of minimal dominating sets in graphs has recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output polynomial time is a fundamental and challenging question; it has been open for several decades and has triggered extensive research. To obtain our result, we present a flipping method to generate all minimal dominating sets of a graph. Its basic idea is to apply a flipping operation to a minimal dominating set $$D^*$$D? to generate minimal dominating sets $$D$$D such that $$G[D]$$G[D] contains more edges than $$G[D^*]$$G[D?]. We show that the flipping method works efficiently on line graphs, resulting in an algorithm with delay $$O(n^2m^2|\mathcal {L}|)$$O(n2m2|L|) between each pair of consecutively output minimal dominating sets, where $$n$$n and $$m$$m are the numbers of vertices and edges of the input graph, respectively, and $$\mathcal {L}$$L is the set of already generated minimal dominating sets. Furthermore, we are able to improve the delay to $$O(n^2m|\mathcal {L}|)$$O(n2m|L|) on line graphs of bipartite graphs. Finally we show that the flipping method is also efficient on graphs of large girth, resulting in an incremental polynomial time algorithm to enumerate the minimal dominating sets of graphs of girth at least 7.

Efficient frequent connected subgraph mining in graphs of bounded tree-width

The frequent connected subgraph mining problem, i.e., the problem of listing all connected graphs that are subgraph isomorphic to at least a certain number of transaction graphs of a database, cannot be solved in output polynomial time in the general case. If, however, the transaction graphs are restricted to forests then the problem becomes tractable. In this paper we generalize the positive result on forests to graphs of bounded tree-width. In particular, we show that for this class of transaction graphs, frequent connected subgraphs can be listed in incremental polynomial time. Since subgraph isomorphism remains NP-complete for bounded tree-width graphs, the positive complexity result of this paper shows that efficient frequent pattern mining is possible even for computationally hard pattern matching operators.

Frequent subgraph mining in outerplanar graphs

In recent years there has been an increased interest in frequent pattern discovery in large databases of graph structured objects. While the frequent connected subgraph mining problem for tree datasets can be solved in incremental polynomial time, it becomes intractable for arbitrary graph databases. Existing approaches have therefore resorted to various heuristic strategies and restrictions of the search space, but have not identified a practically relevant tractable graph class beyond trees. In this paper, we consider the class of outerplanar graphs, a strict generalization of trees, develop a frequent subgraph mining algorithm for outerplanar graphs, and show that it works in incremental polynomial time for the practically relevant subclass of well-behaved outerplanar graphs, i.e., which have only polynomially many simple cycles. We evaluate the algorithm empirically on chemo- and bioinformatics applications.

Incremental polynomial time dualization of quadratic functions and a subclass of degree-k functions

We consider the problem of dualizing a Boolean function f represented by a DNF. In its most general form, this problem is commonly believed not to be solvable by a quasi-polynomial total time algorithm. We show that if the input DNF is quadratic or is a special degree-k DNF, then dualization turns out to be equivalent to hypergraph dualization in hypergraphs of bounded degree and hence it can be achieved in incremental polynomial time.

Learning block-preserving graph patterns and its application to data mining

Recently, due to the rapid growth of electronic data having graph structures such as HTML and XML texts and chemical compounds, many researchers have been interested in data mining and machine learning techniques for finding useful patterns from graph-structured data (graph data). Since graph data contain a huge number of substructures and it tends to be computationally expensive to decide whether or not such data have given structural features, graph mining problems face computational difficulties. Let ${\mathcal{C}}$ be a graph class which satisfies a connected hereditary property and contains infinitely many different biconnected graphs, and for which a special kind of the graph isomorphism problem can be computed in polynomial time. In this paper, we consider learning and mining problems for ${\mathcal{C}}$ . Firstly, we define a new graph pattern, which is called a block preserving graph pattern (bp-graph pattern) for ${\mathcal{C}}$ . Secondly, we present a polynomial time algorithm for deciding whether or not a given bp-graph pattern matches a given graph in ${\mathcal{C}}$ . Thirdly, by giving refinement operators over bp-graph patterns, we present a polynomial time algorithm for finding a minimally generalized bp-graph pattern for ${\mathcal{C}}$ . Outerplanar graphs are planar graphs which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many pharmacologic chemical compounds are known to be represented by outerplanar graphs. The class of connected outerplanar graphs ${\mathcal{O}}$ satisfies the above conditions for ${\mathcal{C}}$ . Next, we propose two incremental polynomial time algorithms for enumerating all frequent bp-graph patterns with respect to a given finite set of graphs in ${\mathcal{O}}$ . Finally, by reporting experimental results obtained by applying the two graph mining algorithms to a subset of the NCI dataset, we evaluate the performance of the two graph mining algorithms.

Listing minimal edge-covers of intersecting families with applications to connectivity problems

Let G = ( V , E ) be a directed/undirected graph, let s , t ∈ V , and let F be an intersecting family on V (that is, X ∩ Y , X ∪ Y ∈ F for any intersecting X , Y ∈ F ) so that s ∈ X and t ∉ X for every X ∈ F . An edge set I ⊆ E is an edge-cover of F if for every X ∈ F there is an edge in I from X to V − X . We show that minimal edge-covers of F can be listed with polynomial delay, provided that, for any I ⊆ E the minimal member of the residual family F I of the sets in F not covered by I can be computed in polynomial time. As an application, we show that minimal undirected Steiner networks, and minimal k -connected and k -outconnected spanning subgraphs of a given directed/undirected graph, can be listed in incremental polynomial time.

Generating 3-vertex connected spanning subgraphs

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O ( K 2 log ( K ) m 2 + K 2 m 3 ) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.

On Enumerating Minimal Dicuts and Strongly Connected Subgraphs

We consider the problems of enumerating all minimal strongly connected subgraphs and all minimal dicuts of a given strongly connected directed graph G=(V,E). We show that the first of these problems can be solved in incremental polynomial time, while the second problem is NP-hard: given a collection of minimal dicuts for G, it is NP-hard to tell whether it can be extended. The latter result implies, in particular, that for a given set of points $\mathcal{A}\subseteq\mathbb{R}^{n}$, it is NP-hard to generate all maximal subsets of $\mathcal{A}$contained in a closed half-space through the origin. We also discuss the enumeration of all minimal subsets of $\mathcal{A}$whose convex hull contains the origin as an interior point, and show that this problem includes as a special case the well-known hypergraph transversal problem.

Generating Cut Conjunctions in Graphs and Related Problems

Let G=(V,E) be an undirected graph, and let B⊆V×V be a collection of vertex pairs. We give an incremental polynomial time algorithm to generate all minimal edge sets X⊆E such that every pair (s,t)∈B of vertices is disconnected in (V,E ∖ X), generalizing well-known efficient algorithms for generating all minimal s-t cuts, for a given pair s,t of vertices. We also present an incremental polynomial time algorithm for generating all minimal subsets X⊆E such that no (s,t)∈B is a bridge in (V,X∪B). Both above problems are special cases of a more general problem that we call generating cut conjunctions for matroids: given a matroid M on ground set S=E∪B, generate all minimal subsets X⊆E such that no element b∈B is spanned by E ∖ X. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V,E∪B), the more general problem of generating cut conjunctions for vectorial matroids turns out to be NP-hard.

On the dualization of hypergraphs with bounded edge-intersections and other related classes of hypergraphs

Given a finite set V , and integers k ≥ 1 and r ≥ 0 , let us denote by A ( k , r ) the class of hypergraphs A ⊆ 2 V with ( k , r ) -bounded intersections, i.e. in which the intersection of any k distinct hyperedges has size at most r . We consider the problem MIS ( A , I ) : given a hypergraph A , and a subfamily I ⊆ I ( A ) of its maximal independent sets (MIS) I ( A ) , either extend this subfamily by constructing a new MIS I ∈ I ( A ) ∖ I or prove that there are no more MIS, that is I = I ( A ) . It is known that, for hypergraphs of bounded dimension A ( 1 , δ ) , as well as for hypergraphs of bounded degree A ( δ , 0 ) (where δ is a constant), problem MIS ( A , I ) can be solved in incremental polynomial time. In this paper, we extend this result to any integers k , r such that k + r = δ is a constant. More precisely, we show that for hypergraphs A ∈ A ( k , r ) with k + r ≤ const , problem MIS ( A , I ) is NC-reducible to the problem MIS ( A ′ , 0̸ ) of generating a single MIS for a partial subhypergraph A ′ of A . In particular, this implies that MIS ( A , I ) is polynomial, and we get an incremental polynomial algorithm for generating all MIS. Furthermore, combining this result with the currently known algorithms for finding a single maximally independent set of a hypergraph, we obtain efficient parallel algorithms for incrementally generating all MIS for hypergraphs in the classes A ( 1 , δ ) , A ( δ , 0 ) , and A ( 2 , 1 ) , where δ is a constant. We also show that, for A ∈ A ( k , r ) , where k + r ≤ const , the problem of generating all MIS of A can be solved in incremental polynomial-time and with space polynomial only in the size of A .

Dual-bounded generating problems: Efficient and inefficient points for discrete probability distributions and sparse boxes for multidimensional data

We show that ∣ X ∣ ≤ n ∣ Y ∣ must hold for two finite sets X , Y ⊂ R n whenever they can be separated by a nonnegative linear function such that X is above Y and the componentwise minimum of any two distinct points in X is dominated by some point in Y . As a consequence, we obtain an incremental quasi-polynomial time algorithm for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p -inefficient points of a given discrete probability distribution, and for generating all maximal hyper-rectangles which contain a specified fraction of points of a given set in R n . This provides a substantial improvement over previously known exponential time algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, implying that for discrete probability distributions with independent coordinates, both p -efficient and p -inefficient points can be separately generated in incremental polynomial time.

ENUMERATING SPANNING AND CONNECTED SUBSETS IN GRAPHS AND MATROIDS(<Special Issue>the 50th Anniversary of the Operations Research Society of Japan)

We show that enumerating all minimal spanning and connected subsets of a given matroid can be solved in incremental quasi-polynomial time. In the special case of graphical matroids, we improve this complexity bound by showing that all minimal 2-vertex connected edge subsets of a given graph can be generated in incremental polynomial time.

Enumerating disjunctions and conjunctions of paths and cuts in reliability theory

Let G = ( V , E ) be a (directed) graph with vertex set V and edge (arc) set E. Given a set P of source–sink pairs of vertices of G, an important problem that arises in the computation of network reliability is the enumeration of minimal subsets of edges (arcs) that connect/disconnect all/at least one of the given source–sink pairs of P . For undirected graphs, we show that the enumeration problems for conjunctions of paths and disjunctions of cuts can be solved in incremental polynomial time. Furthermore, under the assumption that P consists of all pairs within a given vertex set, we also give incremental polynomial time algorithm for enumerating all minimal path disjunctions and cut conjunctions. For directed graphs, the enumeration problem for cut disjunction is known to be NP-complete. We extend this result to path conjunctions and path disjunctions, leaving open the complexity of the enumeration of cut conjunctions. Finally, we give a polynomial delay algorithm for enumerating all minimal sets of arcs connecting two given nodes s 1 and s 2 to, respectively, a given vertex t 1 , and each vertex of a given subset of vertices T 2 .