Algorithms for Variable-Weighted 2-SAT and Dual Problems

  title={Algorithms for Variable-Weighted 2-SAT and Dual Problems},
  author={Stefan Porschen and Ewald Speckenmeyer},
In this paper we study NP-hard variable-weighted satisfiability optimization problems for the class 2-CNF providing worst-case upper time bounds holding for arbitrary real-valued weights. Moreover, we consider the monotone dual class consisting of clause sets where all variables occur at most twice. We show that weighted SAT, XSAT and NAESAT optimization problems for this class are polynomial time solvable using appropriate reductions to specific polynomial time solvable graph problems. 
Complexity Results for Linear XSAT-Problems
It is proved that XSAT remains NP-complete for linear formulas which are monotone and all variables occur exactly l times, and it is shown thatXSAT for this class is NP- complete, in contrast to SAT or NAE-SAT.
XSAT and NAE-SAT of linear CNF classes
On Colored Edge Cuts in Graphs
This work is interested in problems of finding cuts {A,B} which minimize or maximize the number of colors occurring in the edges with exactly one endpoint in A.
Revisitin the enumeration of all models of a Boolean 2-CNF
  • M. Wild
  • Computer Science, Mathematics
  • 2012
An O(Nn^2 + n^2) time algorithm to enumerate all N models of a Boolean 2-CNF with n variables using don't care symbols to show the high efficiency of this method.
Progress on Partial Edge Drawings
A symmetric model (SPED) that requires the two stubs of an edge to be of equal length, and an additional homogeneity constraint that forces the stub lengths to be a given fraction of the edge lengths ($\delta$-SHPED).
Progress on Partial Edge Drawings
An efficient solution for 2-planar drawings and a 2-approximation algorithm for the dual problem are presented and it is shown that, for a fixed stub---edge length ratio δ, not all graphs have a δ-SHPED.
Calculating the Unrooted Subtree Prune-and-Regraft Distance
  • Chris Whidden, F. Matsen
  • Mathematics, Computer Science
    IEEE/ACM Transactions on Computational Biology and Bioinformatics
  • 2019
A “progressive A*” search algorithm is developed using multiple heuristics, including the TBR and replug distances, to exactly compute the unrooted SPR distance, which is nearly two orders of magnitude faster than previous methods on small trees.


Solving Minimum Weight Exact Satisfiability in Time O(20.2441n)
We show that the NP-hard optimization problem minimum weight exact satisfiability for a CNF formula over n propositional variables equipped with arbitrary real-valued weights can be solved in time
On Some Weighted Satisfiability and Graph Problems
This paper provides linear time algorithms solving the optimization problems MINV(MAXV)-NAESAT and MINV (MAXV-XSAT for 2CNF formulas and arbitrary real weights assigned to the variables and considers the relationship between the problems maximum weight independent set and the problem XSAT.
Counting All Solutions of Minimum Weight Exact Satisfiability
We show that the number of all solutions of minimum weight exact satisfiability can be found in O(n2.||C||+20.40567 n) time, for a CNF formula C containing n propositional variables equipped with
Algorithms for Counting 2-SAT Solutions and Colorings with Applications
An algorithm is presented for exactly counting the number of maximum weight satisfying assignments of a 2- Cnf formula that improves on the previous bound of O( 1.246n) by Dahllof, Jonsson, and Wahlstrom.
On variable-weighted exact satisfiability problems
  • Stefan Porschen
  • Mathematics, Computer Science
    Annals of Mathematics and Artificial Intelligence
  • 2007
The algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time and also investigate the corresponding weighted counting problems.
Linear Time Algorithms for Some Not-All-Equal Satisfiability Problems
It is shown that NAESAT can be decided in linear time for monotone formulas in which each clause has length exactly k and each variable occurs exactly k times, and bicolorability of k-uniform, k-regular hypergraphs is decidable inlinear time.
Worst Case Bounds for some NP-Complete Modified Horn-SAT Problems
An exact deterministic algorithm is provided showing that SAT for mixed (hidden) Horn formulas containing n variables is solvable in time O(2) and a fixed-parameter tractability classification for SAT restricted to mixed Horn formulas is obtained.
The complexity of satisfiability problems
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.
Satisfiability of mixed Horn formulas
A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas