Given a set $N$ with $n$ elements and a family $\mathcal{F}$ of subsets, we show how to partition $N$ into $k$ such subsets in $2^n n^{O(1)}$ time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimization versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform. In effect we get exact algorithms in $2^n n^{O(1)}$ time for several well-studied partition problems including domatic number, chromatic number, maximum $k$-cut, bin packing, list coloring, and the chromatic polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering. If only polynomial space is available, our algorithms run in time $3^n n^{O(1)}$ if membership in $\mathcal{F}$ can be decided in polynomial time. We solve chromatic number in $O(2.2461^n)$ time and domatic number in $O(2.8718^n)$ time. Finally, we present a family of polynomial space approximation algorithms that find a number between $\chi(G)$ and $\lceil(1+\epsilon)\chi(G)\rceil$ in time $O(1.2209^n+2.2461^{e^{-\epsilon}n})$.

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Set Partitioning via Inclusion-Exclusion

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions. Thus, one who wishes to obtain useful results from a graph coloring formulation of his problem must do more than just show that the problem is equivalent to the general problem of coloring a graph. If there is to be any hope, one must also obtain information about the structure of the graphs that need to be colored (D.S. Johnson [66]).

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Vertex Colouring and Forbidden Subgraphs – A Survey

The topic of exact, exponential-time algorithms for NP-hard problems has received a lot of attention, particularly with the focus of producing algorithms with stronger theoretical guarantees, e.g. ...

/pdf/algorithms-measures-and-upper-bounds-for-satisfiability-and-55tz1ugj7o.pdf

Algorithms, Measures and Upper Bounds for Satisfiability and Related Problems

Enumerating maximal independent sets with applications to graph colouring

We present a time $\mathcal {O}(1.7548^{n})$algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638n minimal feedback vertex sets and that there exist graphs having 105n/10≈1.5926n minimal feedback vertex sets.

https://www.cs.le.ac.uk/people/ir45/papers/FVSALGORITHMICA.pdf

On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

New algorithms for exact satisfiability

An algorithm for exact satisfiability analysed with the number of clauses as parameter

Inspired by the Maximum Satisfiability and Exact Satisfiability problems we present two Maximum Exact Satisfiability problems. The first problem called Maximum Exact Satisfiability is: given a formula in conjunctive normal form and an integer k, is there an assignment to all variables in the formula such that at least k clauses have exactly one true literal. The second problem called Restricted Maximum Exact Satisfiability has the further restriction that no clause is allowed to have more than one true literal. Both problems are proved NP-complete restricted to the versions where each clause contains at most two literals. In fact Maximum Exact Satisfiability is a generalisation of the well-known NP-complete problem MaxCut. We present an exact algorithm for Maximum Exact Satisfiability where each clause contains at most two literals with time complexity O(poly(L) . 2^{m/4}), where m is the number of clauses and L is the length of the formula. For the second version we give an algorithm with time complexity O(poly(L) . 1.324718^n) , where n is the number of variables. We note that when restricted to the versions where each clause contains exactly two literals and there are no negations both problems are fixed parameter tractable. It is an open question if this is also the case for the general problems.

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Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

The Exact Satisfiability problem is to determine if a CNF-formula has a truth assignment satisfying exactly one literal in each clause; Exact 3-Satisfiability is the version in which each clause contains at most three literals. In this paper, we present algorithms for Exact Satisfiability and Exact 3-Satisfiability running in time O(2^{0.2325n}) and O(2^{0.1379n}), respectively. The previously best algorithms have running times O(2^{0.2441n}) for Exact Satisfiability (Monien, Speckenmeyer and Vornberger (1981)) and O(2^{0.1626n}) for Exact 3-Satisfiability (Kulikov and independently Porschen, Randerath and Speckenmeyer (2002)). We extend the case analyses of these papers and observe, that a formula not satisfying any of our cases has a small number of variables, for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time.

/pdf/new-algorithms-for-exact-satisfiability-2lckggxgj2.pdf

New Algorithms for Exact Satisfiability

We show new lower and upper bounds on the maximum number of maximal induced bipartite subgraphs of graphs with n vertices. We present an infinite family of graphs having 105n-10 ≈ 1.5926n; such subgraphs show an upper bound of O(12n-4) = O(1.8613n) and give an algorithm that finds all maximal induced bipartite subgraphs in time within a polynomial factor of this bound. This algorithm is used in the construction of algorithms for checking k-colorability of a graph. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 127132, 2005

Bolette Ammitzbøll Madsen

Papers

New algorithms for exact satisfiability

An algorithm for exact satisfiability analysed with the number of clauses as parameter

Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

New Algorithms for Exact Satisfiability

On the number of maximal bipartite subgraphs of a graph