P-complete Problem Research Articles

The number of connected sets in Apollonian networks

A vertex subset in a graph that induces a connected subgraph is referred to as a connected set. Counting the number of connected sets N(G) in a graph G is generally a #P-complete problem. In our recent work [Graphs Combin. (2024)], a linear recursive algorithm was designed to count N(G) in any Apollonian network. In this paper we extend our research by establishing a tight upper bound on N(G) in Apollonian networks with an order of n≥3, along with a characterization of the graphs that reach this upper bound. Our approach primarily utilizes linear programming techniques. Moreover, we propose a conjecture regarding the lower bound on N(G) in Apollonian networks with a given order.

Efficient assessment of formal concept stability in the Galois lattice

Formal concept analysis (FCA) is a mathematical tool for analyzing data and formally representing conceptual knowledge. Under this formalism, the concept stability metric can be used in many applications: community detection, pattern selection in noisy data, patient care trajectories, and biclusters of similar objects, etc. However, computing the stability is a #P-complete problem. Therefore, most existing studies have focused either on estimating this metric using some heuristics or by proposing recursive approaches to assess the exact value of the stability. In this paper, we introduce an algorithm called the Exact Stability Computing for Galois Lattice Esc-Gl for computing the stability of a formal concept given the underlying partial relation. Based on the lower cover of a given concept, the algorithm can improve scalability and reduce computational complexity. Our experiments performed on large benchmark datasets from the FCA field show that our contributions sharply outperform all the pioneering approaches of the dedicated literature.

Counting dominating sets in some subclasses of bipartite graphs

Counting dominating sets in a graph is a #P-complete problem even for bipartite graphs. This paper concerns the class of tree convex bipartite graphs, which is a subclass of bipartite graphs. A bipartite graph G with bipartition (X,Y) is called tree convex, if a tree T defined on X exists such that for every vertex y in Y, the neighbors of y form a subtree in T. For each y in Y, when the neighbors of y form a path or a rooted directed path in T, the tree convex bipartite graph is called path-tree bipartite or rooted path-tree bipartite, respectively. This work proves that the problem of counting dominating sets remains #P-complete for path-tree bipartite graphs but admits a polynomial-time solution for the more tightly constrained subclass of rooted path-tree bipartite graphs.

Faster exponential-time algorithms for approximately counting independent sets

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance ε is at most O(20.2680n) times a polynomial in 1/ε. On bipartite graphs, the exponential term in the running time is improved to O(20.2372n). Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, Štefankovič and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless P=NP.

Fast and simple algorithms for counting dominating sets in distance-hereditary graphs

Counting dominating sets (DSs) in a graph is a #P-complete problem even for chordal bipartite graphs and split graphs, which are both subclasses of weakly chordal graphs. This paper investigates this problem for distance-hereditary graphs, which is another known subclass of weakly chordal graphs. This work develops linear-time algorithms for counting DSs and their two variants, total DSs and connected DSs in distance-hereditary graphs.

Counting dominating sets in generalized series-parallel graphs

Counting dominating sets in a graph is a #P-complete problem even in planar graphs. This paper studies this problem for generalized series-parallel graphs, which are a subclass of planar graphs. This work develops some linear-time algorithms for counting dominating sets and their two variants, independent dominating sets and connected dominating sets in generalized series-parallel graphs.

On Decomposition of a Binary Context Without Losing Formal Concepts

The #P-complete problem of finding all the formal concepts of a given context and the decomposition method for its solving are investigated. As parts of the decomposition is proposed to use fragments of the initial context, called boxes. Such decomposition allows to decompose the given context without losing formal concepts and thereby to reduce the execution time of the algorithms for solving considered task. The number of boxes, obtained at each iteration of the decomposition, is determined based on studies of the boxes structure and the rules for stopping of the decomposition process are established

Faster FPTASes for counting and random generation of Knapsack solutions

In the #P-complete problem of counting 0/1 Knapsack solutions, the input consists of a sequence of n nonnegative integer weights w1,…,wn and an integer C, and we have to find the number of subsequences (subsets of indices) with total weight at most C. We give faster and simpler fully polynomial-time approximation schemes (FPTASes) for this problem, and for its random generation counterpart. Our method is based on dynamic programming and discretization of large numbers through floating-point arithmetic. We improve both deterministic counting FPTASes from Gopalan et al. (2011) [9], Štefankovič et al. (2012) [6] and the randomized counting and random generation algorithms in Dyer (2003) [5].Our method is general, and it can be directly applied on top of combinatorial decompositions (such as dynamic programming solutions) of various problems. For example, we also improve the complexity of the problem of counting 0/1 Knapsack solutions in an arc-weighted DAG.

Counting Linear Extensions in Practice: MCMC Versus Exponential Monte Carlo

Counting the linear extensions of a given partial order is a #P-complete problem that arises in numerous applications. For polynomial-time approximation, several Markov chain Monte Carlo schemes have been proposed; however, little is known of their efficiency in practice. This work presents an empirical evaluation of the state-of-the-art schemes and investigates a number of ideas to enhance their performance. In addition, we introduce a novel approximation scheme, adaptive relaxation Monte Carlo (ARMC), that leverages exact exponential-time counting algorithms. We show that approximate counting is feasible up to a few hundred elements on various classes of partial orders, and within this range ARMC typically outperforms the other schemes.

Diving into the queue

We introduce and study the model of diving queue automata which are basically finite automata equipped with a storage medium that is organized as a queue. Additionally, two queue heads are provided at both ends of the queue that can move in a read-only mode inside the queue. In particular, we consider suitable time constraints and the case where only a finite number of turns on the queue is allowed. As one main result we obtain a proper queue head hierarchy, that is, two heads are better than one head, and one head is better than no head. Moreover, it is shown that the model with one queue head, finitely many turns, and no time constraints as well as the model with two queue heads, possibly infinitely many turns, and time constraints is captured by P and has a P-complete membership problem. We obtain also that a subclass of the model with two queue heads is already captured by logarithmic space. Finally, we consider decidability questions and it turns out that almost nothing is decidable for the model with two queue heads, whereas we obtain that at least emptiness and finiteness are decidable for subclasses of the model with one queue head.

Linear-time algorithms for counting independent sets in bipartite permutation graphs

The counting of independent sets in bipartite graphs is a #P-complete problem but counting those in permutation graphs is a problem that can be solved in polynomial-time. This paper develops some linear-time algorithms for counting independent sets, maximal independent sets, and independent perfect dominating sets in the class of bipartite permutation graphs, which is the intersection class between permutation and bipartite graphs.

Competitive self-stabilizing k-clustering

In this paper, we give a silent self-stabilizing algorithm for constructing a k-clustering of any asynchronous connected network with unique IDs. Our algorithm stabilizes in O(n) rounds, using O(log⁡k+log⁡n) space per process, where n is the number of processes. In the general case, our algorithm constructs O(nk)k-clusters. If the network is a Unit Disk Graph (UDG), then our algorithm is 7.2552k+O(1)-competitive, that is, the number of k-clusters constructed by the algorithm is at most 7.2552k+O(1) times the minimum possible number of k-clusters in any k-clustering of the same network. More generally, if the network is an Quasi-Unit Disk Graph (QUDG) with approximation ratio λ, then our algorithm is 7.2552λ2k+O(λ)-competitive. In case of tree networks, our algorithm computes a k-clustering with the minimum number of clusters. Our solution is based on the self-stabilizing construction of a data structure called an MIS tree, a spanning tree of the network whose processes at even levels form a maximal independent set of the network. The MIS tree construction we use (called LFMIS) is the time bottleneck of our k-clustering algorithm, as it takes Θ(n) rounds in the worst case, while the rest of the algorithm takes O(D) rounds, where D is the diameter of the network. We would like to improve that time to be O(D), but we show that our distributed MIS tree construction is a P-complete problem.

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph K⌊n/2⌋,⌈n/2⌉ contains more cycles than any other n-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k. For k=1, we show that any such counterexamples have n≤91 and are not homomorphic to C5; and for any fixed k there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a #P-complete problem in general) in a special case used by our bounds.

Counting independent sets in a tolerance graph

Counting independent sets is a #P-complete problem for general graphs but solvable in polynomial time for interval and permutation graphs. This paper develops some polynomial time algorithms for counting independent sets, maximal independent sets, and independent perfect dominating sets in a tolerance graph, which is a common generalization of interval and permutation graphs. No algorithm for solving those problems for tolerance graphs is currently available.

An Enumerative Algorithm for #2SAT

Counting models for two conjunctive forms (2-CF), problem known as #2SAT, is a classic #P-complete problem. We determine different discrete structures on the constrained graph of the 2-CF formula allowing the efficient computation of #2SAT.We show that if the constrained graph of a 2-CF F is acyclic or it has only cycles, which are independent each other, then #2SAT(F) is computed efficiently. On the other hand, we design a bottom-up procedure to compute #2SAT(F) in an incremental way. Given a formula F, our procedure begins with the maximum subformula which does not have intersecting cycles, let say F0. In each iteration of the procedure, a new clause Ci∈(F−F0) is considered in order to form Fi=(Fi−1∧Ci) and then to compute #2SAT(Fi) based on the computation of #2SAT(Fi−1).

Sublinear-time algorithms for monomer–dimer systems on bounded degree graphs

For a graph G, let Z(G,λ) be the partition function of the monomer–dimer system defined by ∑kmk(G)λk, where mk(G) is the number of matchings of size k in G. We consider graphs of bounded degree and develop a sublinear-time algorithm for estimating log⁡Z(G,λ) at an arbitrary value λ>0 within additive error ϵn with high probability. The query complexity of our algorithm does not depend on the size of G and is polynomial in 1/ϵ, and we also provide a lower bound quadratic in 1/ϵ for this problem. This is the first analysis of a sublinear-time approximation algorithm for a #P-complete problem. Our approach is based on the correlation decay of the Gibbs distribution associated with Z(G,λ). We show that our algorithm approximates the probability for a vertex to be covered by a matching, sampled according to this Gibbs distribution, in a near-optimal sublinear time. We extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability, where again the query complexity is polynomial in 1/ϵ and the lower bound is quadratic in 1/ϵ. Our algorithms are simple to implement and of practical use when dealing with massive datasets. Our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity.

Large scale analysis of signal reachability.

Motivation: Major disorders, such as leukemia, have been shown to alter the transcription of genes. Understanding how gene regulation is affected by such aberrations is of utmost importance. One promising strategy toward this objective is to compute whether signals can reach to the transcription factors through the transcription regulatory network (TRN). Due to the uncertainty of the regulatory interactions, this is a #P-complete problem and thus solving it for very large TRNs remains to be a challenge.Results: We develop a novel and scalable method to compute the probability that a signal originating at any given set of source genes can arrive at any given set of target genes (i.e., transcription factors) when the topology of the underlying signaling network is uncertain. Our method tackles this problem for large networks while providing a provably accurate result. Our method follows a divide-and-conquer strategy. We break down the given network into a sequence of non-overlapping subnetworks such that reachability can be computed autonomously and sequentially on each subnetwork. We represent each interaction using a small polynomial. The product of these polynomials express different scenarios when a signal can or cannot reach to target genes from the source genes. We introduce polynomial collapsing operators for each subnetwork. These operators reduce the size of the resulting polynomial and thus the computational complexity dramatically. We show that our method scales to entire human regulatory networks in only seconds, while the existing methods fail beyond a few tens of genes and interactions. We demonstrate that our method can successfully characterize key reachability characteristics of the entire transcriptions regulatory networks of patients affected by eight different subtypes of leukemia, as well as those from healthy control samples.Availability: All the datasets and code used in this article are available at bioinformatics.cise.ufl.edu/PReach/scalable.htm.Contact: atodor@cise.ufl.eduSupplementary information: Supplementary data are available at Bioinformatics online.

Tight Bounds for Monotone Switching Networks via Fourier Analysis

We prove tight size bounds on monotone switching networks for the NP-complete problem of $k$-clique, and for an explicit monotone problem by analyzing a pyramid structure of height $h$ for the P-complete problem of generation. This gives alternative proofs of the separations of m-NC from m-P and of m-NC$^i$ from m-NC$^{i+1}$, different from Raz--McKenzie (Combinatorica 1999). The enumerative-combinatorial and Fourier analytic techniques in this paper are very different from a large body of work on circuit depth lower bounds, and may be of independent interest. An earlier version of this paper appeared in the Proceedings of the 44th ACM Symp. on Theory of Computing, pp. 495-504, 2011.

Application of permanents of square matrices for DNA identification in multiple-fatality cases

BackgroundDNA profiling is essential for individual identification. In forensic medicine, the likelihood ratio (LR) is commonly used to identify individuals. The LR is calculated by comparing two hypotheses for the sample DNA: that the sample DNA is identical or related to a reference DNA, and that it is randomly sampled from a population. For multiple-fatality cases, however, identification should be considered as an assignment problem, and a particular sample and reference pair should therefore be compared with other possibilities conditional on the entire dataset.ResultsWe developed a new method to compute the probability via permanents of square matrices of nonnegative entries. As the exact permanent is known as a #P-complete problem, we applied the Huber–Law algorithm to approximate the permanents. We performed a computer simulation to evaluate the performance of our method via receiver operating characteristic curve analysis compared with LR under the assumption of a closed incident. Differences between the two methods were well demonstrated when references provided neither obligate alleles nor impossible alleles. The new method exhibited higher sensitivity (0.188 vs. 0.055) at a threshold value of 0.999, at which specificity was 1, and it exhibited higher area under a receiver operating characteristic curve (0.990 vs. 0.959, P = 9.6E-15).ConclusionsOur method therefore offers a solution for a computationally intensive assignment problem and may be a viable alternative to LR-based identification for closed-incident multiple-fatality cases.

Approximations of the Generalized Cascade Model

The study of infection processes is an important field of science both from the theoretical and the practical point of view, and has many applications. In this paper we focus on the popular Independent Cascade model and its generalization. Unfortunately the exact computation of infection probabilities is a #P-complete problem [8], so one cannot expect fast exact algorithms. We propose several methods to efficiently compute infection patterns with acceptable accuracy. We will also examine the possibility of substituting the Independent Cascade model with a computationally more tractable model.