Statement and solution of some problems on a mathematical safe

The problem on a mathematical safe is formulated and studied in terms of graph theory. The cases of simplest digraphs such as paths, contours, and doubly connected components are analyzed. A number of statements on the existence of solutions to these problems are proved. The results obtained are extended to the case of corresponding nondirected graphs. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 3–14, May–June 2006.     

Solution of a Matrix Problem on a Mathematical Safe for Locks of the Same Kind. I

A mathematical safe problem is first formulated in a general form. The case of a set of locks that are of the same kind and are arranged in the form of a rectangular matrix is considered. Locks are considered to be of the same kind if their states are specified by numbers from the class of residues to a definite modulus. In the case of a prime modulus, a complete solution is provided. All the results obtained are illustrated by examples.__________Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 155–167, March–April 2005.     

Formal verification of concurrent programs with read-write locks

Read-write locking is an important mechanism to improve concurrent granularity, but it is difficult to reason about the safety of concurrent programs with read-write locks. Concurrent separation logic (CSL) provides a simple but powerful technique for locally reasoning about concurrent programs with mutual exclusive locks. Unfortunately, CSL cannot be directly applied to reasoning about concurrent programs with read-write locks due to the different concurrent control mechanisms.This paper focuses on extending CSL and presenting a proof-carrying code (PCC) system for reasoning about concurrent programs with read-write locks. We extend the heap model with a writing permission set, denoted as logical heap, then define “strong separation” and “weak separation” over logical heap. Following CSL’s local-reasoning idea, we develop a novel program logic to enforce weak separations of heap between different threads and provide verification of concurrent programs with read-write locks.     

LP-oriented upper bounds for the weighted stability number of a graph

Upper bounds for the weighted stability number of a graph are considered that are based on the approximation of its stable set polytope by linear inequalities for odd cycles and p-wheels in the graph. Algorithms are developed for finding upper bounds on the basis of solution of LP problems with a finite number of inequalities produced by the shortest path algorithm for a special graph. The results of test experiments are given for graphs with several hundred or thousand vertices. This work was partially financially supported by the CRDF Cooperative Grants Program under grant UKM2-2812-KV-06. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 157–170, January–February 2009.     

Adaptive locks: Combining transactions and locks for efficient concurrency

Transactional memory is being advanced as an alternative to traditional lock-based synchronization for concurrent programming. Transactional memory simplifies the programming model and maximizes concurrency. At the same time, transactions can suffer from interference that causes them to often abort, from heavy overheads for memory accesses, and from expressiveness limitations (e.g., for I/O operations). In this paper we propose an adaptive locking technique that dynamically observes whether a critical section would be best executed transactionally or while holding a mutex lock. The critical new elements of our approach include the adaptivity logic and cost–benefit analysis, a low-overhead implementation of statistics collection and adaptive locking in a full C compiler, and an exposition of the effects on the programming model. In experiments with both micro and macrobenchmarks we found adaptive locks to consistently match or outperform the better of the two component mechanisms (mutexes or transactions). Compared to either mechanism alone, adaptive locks often provide 3-to-10x speedups. Additionally, adaptive locks simplify the programming model by reducing the need for fine-grained locking: with adaptive locks, the programmer can specify coarse-grained locking annotations and often achieve fine-grained locking performance due to the transactional memory mechanisms.     

Type‐safe concurrent resource sharing

Concurrent systems often have many processes sharing a common set of resources, both memory regions and hardware devices. Among the many challenges in producing safe concurrent software are single access, atomic transactions, starvation, and deadlock. Locks are frequently used to provide single access to shared resources, but do not guarantee safe usage. This paper extends the previous work on linear, singleton, and arithmetic types and linear memory primitives. Our contributions are capabilities for shared resources, and locks to control these capabilities in provably safe ways. We present formalized locks in a lambda calculus along with the soundness properties of preservation and progress. The type system described here prevents data races. The formalized locks have also been implemented in a C‐like language and used in a network device driver. Copyright © 2010 John Wiley & Sons, Ltd.     

Algorithms for solution of systems of linear diophantine equations in residue fields

Algorithms are proposed for computing the basis of the solution set of a system of linear Diophantine homogeneous or inhomogeneous equations in the residue field modulo a prime number. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 15–23, March–April 2007.     

Inference of structures of models of probabilistic dependences from statistical data

Problems of reconstruction of structures of probabilistic dependence models in the class of directed (oriented) acyclic graphs (DAGs) and mono-flow graphs are considered. (Mono-flow graphs form a subclass of DAGs in which the cycles with one collider are prohibited.) The technique of induced (provoked) dependences is investigated and its application to the identification of structures of models is shown. The algorithm “Collifinder-M” is developed that identifies all collider variables (i.e., solves an intermediate problem of reconstruction of the structure of a mono-flow model). It is shown that a generalization of the technique of induced dependences makes it possible to strengthen well-known rules of identification of orientation of edges in a DAG model. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 19–31, November–December 2005.     

Interval-parameter optimization problems on graphs

Well-known optimization problems on graphs are considered under uncertainty when parameter domains are specified in the form of intervals. Exponential estimates of computational complexity of problems being studied and also problems that are polynomial in the classical formulation are substantiated. Polynomially solvable subclasses are found, and sufficient conditions of statistical efficiency of a proposed approximate algorithm are constructively substantiated. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 3–14, March–April 2009.     

Invariance theorems for a class of systems of random nonlinear equations over an arbitrary finite ring with left unity

A class of homogenous systems of random nonlinear equations over an arbitrary finite ring with left unity is considered. The author analyzes the invariance boundaries for limit factorial moments of nonzero solutions, the limit distribution of the number of nonzero solutions, and the geometrical structure of the set of nonzero solutions of the system as the number of unknowns tends to infinity. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 106–115, November–December 2008.     

Solution of continuous problems of optimal covering with spheres using optimal set-partition theory

The paper considers a continuous problem of optimal c-sphere covering of a compact set from Ω from E _n with a given number of spheres of minimum radius and a problem of covering a set with the minimum number of spheres of given radius. Algorithms are proposed and substantiated to solve the problems using optimal set-partition theory and Shor’s r-algorithm. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 98–117, May–June 2009.     

Exact Sampling from Perfect Matchings of Dense Regular Bipartite Graphs

We present the first algorithm for generating random variates exactly uniformly from the set of perfect matchings of a bipartite graph with a polynomial expected running time over a nontrivial set of graphs. Previous Markov chain results obtain approximately uniform variates for arbitrary graphs in polynomial time, but their general running time is Θ(n¹⁰ (ln n)²). Other algorithms (such as Kasteleyn's O(n³) algorithm for planar graphs) concentrated on restricted versions of the problem. Our algorithm employs acceptance/rejection together with a new upper limit on the permanent of a form similar to Bregman's theorem. For graphs with 2n nodes, where the degree of every node is γn for a constant γ, the expected running time is O(n^{1.5 + .5/γ}). Under these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can generate approximately uniform variates in Θ(n^{4.5 + .5/γ} ln n) time, making our algorithm significantly faster on this class of graphs. The problem of counting the number of perfect matchings in these types of graphs is # P complete. In addition, we give a 1 + σ approximation algorithm for finding the permanent of 0–1 matrices with identical row and column sums that runs in O(n^{1.5 + .5/γ} (1/σ²) log (1/δ))$, where the probability that the output is within 1 + \sigma$ of the permanent is at least 1 – δ.     

Space-Efficient Counting in Graphs on Surfaces

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2^k, for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. The techniques used are algebraic topological that may be useful in many other problems involving planar or higher genus graphs – such as higher genus graph recognition in Logspace. In the spirit of counting problems modulo 2^k, we also exhibit a highly parallel ?L\oplus {\bf L} algorithm for finding the value of a permanent modulo 2^k. Previously, the best known result in this direction was Valiant’s result that this problem lies in P. We also show that we can count the number of perfect matchings modulo 2^k in an arbitrary graph in P. This extends Valiant’s result for the permanent, since the Permanent may be modeled as counting the number of perfect matchings in bipartite graphs.     

Isomorphism of regular NM-graphs of degree 4

An analysis of the problem of isomorphism of natural modular graphs is continued. New results are obtained for regular graphs of degree 4. The general approach to the analysis of arbitrary regular NM-graphs is developed, which brings close to solving the isomorphism problem for a given class of numerical graphs. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 95–103, January–February 2006.     

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph. A preliminary version of this paper appeared in the Proceedings of 10th Scandinavian Workshop on Algorithm Theory (SWAT), Lecture Notes in Computer Science, vol. 4059, pp. 304–315, 2006.     

Properties of the DGS-Auction Algorithm

This paper investigates algorithmic properties and overall performance of the exact auction algorithm in Demange, Gale and Sotomayor (J. Polit. Economy 94: 863–872, 1986) or DGS for short. This task is achieved by interpreting DGS as a graph and by conducting a large number of computer simulations. The crucial step in DGS is when the auctioneer selects a so-called minimal overdemanded set of items because the specific selection may affect a number of performance measures such as the number of iterations and the ratio of elicited preferences. The computational results show that (i) DGS graphs are typically large even for relatively small numbers of bidders and items, (ii) DGS converges slowly and (iii) DGS performs well in terms of preference elicitation. The paper also demonstrates that the modification to DGS based on the Ford–Fulkerson algorithm outperforms all investigated rules for selecting a minimal overdemanded set of items in DGS both in terms of termination speed and preference elicitation.     

Boolean problem of sequential minimization of moduli of linear functions and stability theorems

A multicriterion Boolean problem of lexicographic optimization with partial criteria represented by moduli (absolute values) of linear functions is considered. Five types of stability of a set of lexicographic optima under “small” variations in the parameters of a vector criterion are investigated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 178–187, September–October 2007.     

Laplacian spectrum analysis and spanning tree algorithm for circuit partitioning problems

The spectrum of a graph is the set of all eigenvalues of the Laplacian matrix of the graph. There is a closed relationship between the Laplacian spectrum of graphs and some properties of graphs such as connectivity. In the recent years Laplacian spectrum of graphs has been widely applied in many fields. The application of Laplacian spectrum of graphs to circuit partitioning problems is reviewed in this paper. A new criterion of circuit partitioning is proposed and the bounds of the partition ratio for weighted graphs are also presented. Moreover, the deficiency of graph-partitioning algorithms by Laplacian eigenvectors is addressed and an algorithm by means of the minimal spanning tree of a graph is proposed. By virtue of taking the graph structure into consideration this algorithm can fulfill general requirements of circuit partitioning.     

Categorical approach to the construction of fuzzy graph grammars

The categorical approach is proposed to the formalization of fuzzy graph grammars obtained as a result of generalization of sequential graph grammars. This approach takes into consideration the basic types of fuzziness that arise in constructing categories of fuzzy objects and describing transformations of fuzzy graphs generated by fuzzy sets. All the problems of undecidability that are well known for Chomsky grammars are proved to hold true for fuzzy graph grammars. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 130–144, July–August 2006.     

Algorithms for solving systems of linear diophantine equations in residue rings

Algorithms are proposed that construct the basis of the set of solutions to a system of homogeneous or inhomogeneous linear Diophantine equations in a residue ring modulo n when the prime factors of n are known. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 27–40, November–December 2007.