P-complete Problem Research Articles

Complexity of approximating the vertex centroid of a polyhedron

Let P be an H-polytope in Rd with vertex set V. The vertex centroid is defined as the average of the vertices in V. We first prove that computing the vertex centroid of anH-polytope, or even just checking whether it lies in a given halfspace, is #P-hard. We also consider the problem of approximating the vertex centroid by finding a point within an ϵ distance from it and prove this problem to be #P-easy in the sense that it can be solved efficiently using an oracle for some #P-complete problem. In particular, we show that given an oracle for counting the number of vertices of an H-polytope, one can approximate the vertex centroid in polynomial time. Counting the number of vertices of a polytope defined as the intersection of halfspaces is known to be #P-complete. We also show that any algorithm approximating the vertex centroid to any “sufficiently” non-trivial (for example constant) distance, can be used to construct a fully polynomial-time approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid cannot be approximated to a distance of d12−δ for any fixed constant δ>0 unless P=NP.

Counting Independent Sets Using the Bethe Approximation

We consider the #P-complete problem of counting the number of independent sets in a given graph. Our interest is in understanding the effectiveness of the popular belief propagation (BP) heuristic. BP is a simple iterative algorithm that is known to have at least one fixed point, where each fixed point corresponds to a stationary point of the Bethe free energy (introduced by Yedidia, Freeman, and Weiss [IEEE Trans. Inform. Theory, 51 (2004), pp. 2282–2312] in recognition of Bethe’s earlier work in 1935). The evaluation of the Bethe free energy at such a stationary point (or BP fixed point) leads to the Bethe approximation for the number of independent sets of the given graph. BP is not known to converge in general, nor is an efficient, convergent procedure for finding stationary points of the Bethe free energy known. Furthermore, the effectiveness of the Bethe approximation is not well understood. As the first result of this paper we propose a BP-like algorithm that always converges to a stationary point ...

A simple P-complete problem and its language-theoretic representations

A variant of the Circuit Value Problem is introduced, in which every gate implements the NOR function ¬ ( x ∨ y ) , and one of the inputs of every k th gate must be the ( k − 1 ) th gate. The problem, which remains P-complete, is encoded as a simple formal language over a two-letter alphabet, which can be succinctly represented by language equations of several types. Using this representation, a conjunctive grammar with 8 rules, a Boolean grammar with 5 rules and an LL(1) Boolean grammar with 8 rules for this language are constructed. Another encoding of the problem is represented by a trellis automaton with 11 states and a linear conjunctive grammar with 20 rules.

Estimating the relevance on Communication Lines Based on the Number of Edge Covers

Abstract Counting the number of edge covers on graphs, denoted as the #Edge_Covers problem, is a #P-complete problem. Knowing the number of edge covers is useful for estimating the relevance of the lines in a communication network, which is an important measure in the reliability analysis of a network. In this paper, we present efficient algorithms for solving the #Edge_Covers problem on the most common network topologies, namely, Bus, Stars, Trees and Rings. We show that if the topology of the network G does not contain intersecting cycles (any pair of cycles with common edges), then the number of edge covers can be computed in linear time on the size of G.

Optimal Value of Information in Graphical Models

Many real-world decision making tasks require us to choose among several expensive observations. In a sensor network, for example, it is important to select the subset of sensors that is expected to provide the strongest reduction in uncertainty. In medical decision making tasks, one needs to select which tests to administer before deciding on the most effective treatment. It has been general practice to use heuristic-guided procedures for selecting observations. In this paper, we present the first efficient optimal algorithms for selecting observations for a class of probabilistic graphical models. For example, our algorithms allow to optimally label hidden variables in Hidden Markov Models (HMMs). We provide results for both selecting the optimal subset of observations, and for obtaining an optimal conditional observation plan. Furthermore we prove a surprising result: In most graphical models tasks, if one designs an efficient algorithm for chain graphs, such as HMMs, this procedure can be generalized to polytree graphical models. We prove that the optimizing value of information is $NP^{PP}$-hard even for polytrees. It also follows from our results that just computing decision theoretic value of information objective functions, which are commonly used in practice, is a #P-complete problem even on Naive Bayes models (a simple special case of polytrees). In addition, we consider several extensions, such as using our algorithms for scheduling observation selection for multiple sensors. We demonstrate the effectiveness of our approach on several real-world datasets, including a prototype sensor network deployment for energy conservation in buildings.

Representing a P-complete problem by small trellis automata

A restricted case of the Circuit Value Problem known as the Sequential NOR Circuit Value Problem was recently used to obtain very succinct examples of conjunctive grammars, Boolean grammars and language equations representing P-complete languages (Okhotin, http://dx.doi.org/10.1007/978-3-540-74593-8_23 "A simple P-complete problem and its representations by language equations", MCU 2007). In this paper, a new encoding of the same problem is proposed, and a trellis automaton (one-way real-time cellular automaton) with 11 states solving this problem is constructed.

Holographic Algorithms

Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in one-to-one, or possibly one-to-many, fashion. In this paper we propose a new kind of reduction that allows for gadgets with many-to-many correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a P-complete problem, then polynomial systems can be obtained, the solvability of which would imply P$^{{\tiny{\#P}}}$ = NC2.

Some decision and counting problems of the Duquenne–Guigues basis of implications

Implications of a formal context obey Armstrong rules, which allows one to define a minimal (in the number of implications) implication basis, called Duquenne–Guigues basis or stem base in the literature. In this paper we show how implications are reduced to functional dependencies and prove that the problem of determining the size of the stem base is a #P-complete problem.

Algorithmes parallèles à grain adaptatif et applications

To tune granularity of tasks at runtime, we introduce an original algorithmic scheme. It is based on the coupling of two algorithms, one sequential, the other one parallel and fine grain. However, parallelism is generated only in case of idleness of some processor. Then, when executing the program on a limited number of resources, the overhead related to parallelism management is bounded with no restriction of potential parallelism. It is especially suited to applications for which parallelization drastically increases the number of operations or induces a loss of performance, despite a decreasing execution time. This scheme is applied to parallelisation of two applications: gzip (Gailly, 2003) that implements Lempel-Ziv compression, a P-complete problem; and PL (ProBayes, n.d.) a probabilistic inference engine.

A hybrid algorithm for computing permanents of sparse matrices

The permanent of matrices has wide applications in many fields of science and engineering. It is, however, a #P-complete problem in counting. The best-known algorithm for computing the permanent, which is due to Ryser [Combinatorial Mathematics, The Carus Mathematical Monographs, vol. 14, Mathematical Association of America, Washington, DC, 1963], runs O(n2^n^-^1) in time. It is possible to speed up algorithms for matrices with special structures, which arise commonly in applications. Most algorithms discussed before focus on 0,1 matrix. In this paper, a hybrid algorithm is proposed. It is efficient for sparse matrices.

On the probability of rendezvous in graphs

In a simple graph G without isolated nodes the following random experiment is carried out: each node chooses one of its neighbors uniformly at random. We say a rendezvous occurs if there are adjacent nodes u and v such that u chooses v and v chooses u; the probability that this happens is denoted by s(G). Métivier, Saheb, and Zemmari [“Randomized rendezvous,” Algorithms, trees, combinatorics, and probabilities, Eds. G. Gardy and A. Mokkadem, Trends in Mathematics, 183–194. Birkhäuser, Boston, 2000.] asked whether it is true that s(G) ≥ s(Kn) for all n-node graphs G, where Kn is the complete graph on n nodes. We show that this is the case. Moreover, we show that evaluating s(G) for a given graph G is a #P-complete problem, even if only d-regular graphs are considered, for any d ≥ 5. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005

Algorithms for four variants of the exact satisfiability problem

We present four polynomial space and exponential time algorithms for variants of the E XACT S ATISFIABILITY problem. First, an O(1.1120 n ) (where n is the number of variables) time algorithm for the NP-complete decision problem of E XACT 3-S ATISFIABILITY , and then an O(1.1907 n ) time algorithm for the general decision problem of E XACT S ATISFIABILITY . The best previous algorithms run in O(1.1193 n ) and O(1.2299 n ) time, respectively. For the #P-complete problem of counting the number of models for E XACT 3-S ATISFIABILITY we present an O(1.1487 n ) time algorithm. We also present an O(1.2190 n ) time algorithm for the general problem of counting the number of models for E XACT S ATISFIABILITY ; presenting a simple reduction, we show how this algorithm can be used for computing the permanent of a 0/1 matrix.

Algorithms for four variants of the exact satisfiability problem

We present four polynomial space and exponential time algorithms for variants of the E XACT S ATISFIABILITY problem. First, an O(1.1120 n ) (where n is the number of variables) time algorithm for the NP-complete decision problem of E XACT 3-S ATISFIABILITY, and then an O(1.1907 n ) time algorithm for the general decision problem of E XACT S ATISFIABILITY. The best previous algorithms run in O(1.1193 n ) and O(1.2299 n ) time, respectively. For the #P-complete problem of counting the number of models for E XACT 3-S ATISFIABILITY we present an O(1.1487 n ) time algorithm. We also present an O(1.2190 n ) time algorithm for the general problem of counting the number of models for E XACT S ATISFIABILITY; presenting a simple reduction, we show how this algorithm can be used for computing the permanent of a 0/1 matrix.

Using Program Schemes to Capture Polynomial-Time Logically on Certain Classes of Structures

AbstractIn this paper, the study of the expressive power of certain classes of program schemes on finite structures is continued, in relation to more mainstream logics studied in finite model theory and to computational complexity. The author shows that there exists a program scheme – whose constructs are assignments and while-loops with quantifier-free tests and which has access to a stack – that can accept a P-complete problem, the deterministic path system problem, even in the absence of non-determinism, so long as problem instances are presented in a functional style. (The proof given here leans heavily on Cook's proof that the classes of formal languages accepted by deterministic and non-deterministic logspace auxiliary pushdown machines coincide.) However, whilst this result is of independent interest, in that it leads to a deterministic model of computation capturing P, whose non-deterministic variant also captures P, the program scheme can also be used to build a successor relation in certain classes of structures (namely: the class of strongly connected locally ordered digraphs, the class of connected planar embeddings, and the class of triangulations), with the consequence that on these classes of graphs, (a fragment of) path system logic (with no built-in relations) captures exactly the polynomial-time solvable problems.

The Linkage of a Graph

The linkage of a graph is defined to be the maximum min-degree of any of its subgraphs. It is known that the linkage of a graph is equal to its width: for an arbitrary linear ordering of the vertices of the graph, consider the maximum, with respect to any vertex $v$, of the number of vertices connected with $v$ and preceding it in the ordering; the width of the graph is the minimum of these maxima over all possible linear orderings. Width has been used in artificial intelligence in the context of constraint satisfaction problems (CSPs). A more general notion is defined by considering not the number of vertices preceding and connected with $v$ but rather the least number of vertices preceding and connected with any cluster of at most $j$ consecutive vertices extending to the right up to $v$ ($j$ is a given integer). The graph parameter thus defined is called $j$-width. No efficient algorithm was known for computing the $j$-width. In this paper, we introduce a graph parameter depending on $j$ that refers to the subgraphs of the graph and generalizes the notion of linkage. We prove the min--max theorem that this graph parameter, which we call $j$-linkage, is equal to $j$-width, and we then give a polynomial-time algorithm for computing it (for constant $j$). We also find tight lower and upper bounds for the $j$-linkage (equivalently, the $j$-width) of graphs with given numbers of vertices and edges. It is interesting to note that a lower bound for the width of a graph had been found by Erdos; as we show, however, that bound is not tight. Moreover, we prove that our lower bound for width is also a tight lower bound for treewidth, pathwidth, and bandwidth, graph parameters that may be arbitrarily larger than width. Finally, we show that computing the $j$-linkage is a P-complete problem, whereas we prove that approximating it is a threshold problem: it is in NC for approximation factors $ 1/2$.

The Complexity of Counting Problems in Equational Matching

We introduce a class of counting problems that arise naturally in equational matching and investigate their computational complexity. If E is an equational theory, then #E-Matching is the problem of counting the number of most general E-matchers of two given terms. #E-Matching is a well-defined algorithmic problem for every finitary equational theory. Moreover, it captures more accurately the computational difficulties associated with finding minimal complete sets of E-matchers than the corresponding decision problem for E-matching does. In 1979, L. Valiant developed a computational model for measuring the complexity of counting problems and demonstrated the existence of #P- complete problems, i.e., counting problems that are complete for counting non-deterministic Turing machines of polynomial-time complexity. Using the theory of #P-completeness, we analyze the computational complexity of #E-matching for several important equational theories E. We establish that if E is one of the equational theories A, C, AC, I, U, ACI, Set, ACU, or ACIU, then #E-Matching is a #P-complete problem. We also show that there are equational theories, such as the restriction of AC-matching to linear terms, for which the underlying decision matching problem is solvable in polynomial time, while the associated counting matching problem is #P-complete.

A loop-free algorithm for generating the linear extensions of a poset

A precise concept of when a combinatorial counting problem is “hard” was first introduced by Valiant (1979) when he defined the notion of a #P-complete problem. Correspondingly, there has been consistent interest in the notion of when a combinatorial listing problem admits a very special regular structure in which transition times between objects being listed are uniformly bounded by a fixed constant. Early descriptions of suchloop-free listing algorithms may be found in the bookAlgorithmic Combinatorics by Even (1973). Recently, the problem of counting all linear extensions of a partially ordered set has received attention with regard to both of these combinatorial concepts. Brightwell and Winkler (1991) have shown, by a very ingenious argument, that the poset-extension counting problem is #P-complete. Pruesse and Ruskey (1992) have shown that the corresponding listing problem can be solved in constant amortized time and have posed the problem of finding a loop-free algorithm for the poset-extension problem. The present paper presents a solution to this latter problem. This sequence of results represents an interesting juxtaposition, in a fixed, naturally-occurring combinatorial problem, of intricate and precisely defined “irregularities” with respect to counting with very strong regularities with respect to listing.

Computational complexity of loss networks

In this paper we examine the theoretical limits on developing algorithms to find blocking probabilities in a general loss network. We demonstrate that exactly computing the blocking probabilities of a loss network is a # P-complete problem. We also show that a general algorithm for approximating the blocking probabilities is also intractable unless RP = NP, which seems unlikely according to current common notions in complexity theory. Given these results, we examine implications for designing practical algorithms for finding blocking probabilities in special cases.

A note on a counting problem arising in percolation theory

We investigate an esoteric enumeration problem on graphs which arises from a physical model related to the continuum percolation theory. It counts signed objects, but surprisingly, we are able to find an equivalent count of positive signs only, namely, restricted acyclic orientations. This reduces the counting effort, but shows on the other hand that it is a #P-complete problem.

The complexity of membership for deterministic growing context-sensitive grammars∗

In this note, we present an interesting P-complete problem concerning growing context-sensitive grammars, a proper subclass of the context-sensitive grammars. We show that uniform membership for forward deterministic growing context-sensitive grammars is P-complete. As a consequence, we obtain a simple proof for the NP-completeness of the uniform membership problem for growing context- sensitive grammars.