Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

Abstract

We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusionexclusion characterizations.We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m l O(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2 n n O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732 n ) and exponential space.We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three.We extend the analysis to a number of related problems such as TSP and Chromatic Number.

Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG

“…The usual trick is to apply divide and conquer paradigm to reduce the space requirement. It has been used in [11,1] for other problems.…”

“…For decision problems with input size n, and a parameter k (which typically, and in all the problems we consider in this paper, is the solution size), the goal in parameterized algorithms is to design an exact algorithm with runtime f (k)n O (1) where f is a function of k alone, against a trivial n k+O (1) algorithm. Problems having such an algorithm is said to be fixed parameter tractable (FPT), and such algorithms are practical when small parameters cover practical ranges.…”

“…Recent years have seen a lot of growing interest in the field and have led to the development of fast exponential time algorithms for various problems, including SATISFIABILITY [12,22], COLOR-ING [4,2,1,20], MAXIMUM INDEPENDENT SET [21,6], DOMINATING SET [8,9], ODD CYCLE TRANSVERSAL [2,20] and many others. See recent surveys by Fomin et al [7] and Woeginger [23] for an overview.…”

Exponential Time Complexity of the Permanent and the Tutte Polynomial

“…Such algorithms are also known for general graphs [BH08], the current best bound is O(1.619 n ) [Koi09].…”

Monitoring distributed computing systems on the basis of the determined shortest paths and shortest hamiltonian cycles in a graph

“…В них для решения ЗК предложен алгоритм динамического программирования, имею-щий временную сложность O(2 n ) и использующий память M (M -максимальный вес ребра). Лучший алгоритм с полиномиальной памятью с временной сложностью O(4 n n log n ) получен в работах [9,10]. В ра-боте [11] показано, что ЗК может быть решена за время O(2 2n-t n log(n-t) ) и памятью О(2 t ) для любого t=n, n/2, n/4, … В работах [12][13][14][15] …”