17,397 research outputs found
On a variant of Monotone NAE-3SAT and the Triangle-Free Cut problem
In this paper we define a restricted version of Monotone NAE-3SAT and show
that it remains NP-Complete even under that restriction. We expect this result
would be useful in proving NP-Completeness results for problems on
-colourable graphs (). We also prove the NP-Completeness of the
Triangle-Free Cut problem
Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT
In directed graphs, we investigate the problems of finding: 1) a minimum
feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2)
a feedback vertex set inducing an acyclic graph (also called the Vertex
2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a
minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS).
We show that these problems are strongly related to (variants of) Monotone
3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in
positive form. As a consequence, we deduce several NP-completeness results on
restricted versions of these problems. In particular, we define the 2-Choice
version of an optimization problem to be its restriction where the optimum
value is known to be either D or D+1 for some integer D, and the problem is
reduced to decide which of D or D+1 is the optimum value. We show that the
2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones
Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of
Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth
assignment minimize the number of variables set to true.
Finally, we propose two classes of directed graphs for which Acyclic FVS is
polynomially solvable, namely flow reducible graphs (for which MFVS is already
known to be polynomially solvable) and C1P-digraphs (defined by an adjacency
matrix with the Consecutive Ones Property)
FPTAS for Counting Monotone CNF
A monotone CNF formula is a Boolean formula in conjunctive normal form where
each variable appears positively. We design a deterministic fully
polynomial-time approximation scheme (FPTAS) for counting the number of
satisfying assignments for a given monotone CNF formula when each variable
appears in at most clauses. Equivalently, this is also an FPTAS for
counting set covers where each set contains at most elements. If we allow
variables to appear in a maximum of clauses (or sets to contain
elements), it is NP-hard to approximate it. Thus, this gives a complete
understanding of the approximability of counting for monotone CNF formulas. It
is also an important step towards a complete characterization of the
approximability for all bounded degree Boolean #CSP problems. In addition, we
study the hypergraph matching problem, which arises naturally towards a
complete classification of bounded degree Boolean #CSP problems, and show an
FPTAS for counting 3D matchings of hypergraphs with maximum degree .
Our main technique is correlation decay, a powerful tool to design
deterministic FPTAS for counting problems defined by local constraints among a
number of variables. All previous uses of this design technique fall into two
categories: each constraint involves at most two variables, such as independent
set, coloring, and spin systems in general; or each variable appears in at most
two constraints, such as matching, edge cover, and holant problem in general.
The CNF problems studied here have more complicated structures than these
problems and require new design and proof techniques. As it turns out, the
technique we developed for the CNF problem also works for the hypergraph
matching problem. We believe that it may also find applications in other CSP or
more general counting problems.Comment: 24 pages, 2 figures. version 1=>2: minor edits, highlighted the
picture of set cover/packing, and an implication of our previous result in 3D
matchin
Bonyolultságelmélet és algebra = Complexity and algebra
Az elmĂşlt nĂ©gy Ă©vben számos, a CSP-vel kapcsolatos problĂ©mát vizsgáltunk. 43 tudományos dolgozatot publikáltunk, ezek nagy rĂ©szĂ©t vezetĹ‘ nemzetközi folyĂłiratokban. Legfontosabb eredmĂ©nyeink a következĹ‘k: A Feder-Vardi tĂ©telt általánosĂtva beláttuk, hogy a CSP problĂ©mák osztálya polinomiálisan ekvivalens a Monotone Monadic Strict NP osztállyal. Igazoltuk, hogy a lineáris programmal approximálhatĂł CSP problĂ©mák osztálya pontosan az 1-szĂ©lessĂ©gű osztály. BebizonyĂtottuk, hogy fĂĽggvĂ©nyteljes algebrákra az ekvivalenciaproblĂ©ma bonyolultsága coNP-teljes. Továbbá beláttuk, hogy kommutatĂv gyűrűk felett a szigma ekvivalencia problĂ©ma P-beli, ha pedig a gyűrű Jacobson-radikál szerinti faktora nemkommutatĂv, akkor coNP-teljes. KezdemĂ©nyeztĂĽk a kiterjesztett egyenletmegoldhatĂłság Ă©s ekvivalencia vizsgálatát, Ă©s beláttuk a kapcsolĂłdĂł dichotĂłmiatĂ©teleket csoportokra. | In the last four years we studied several problems concerning CSP. We published 43 research papers, mainly in leading international journals. Our most important results are the following: We generalized the Feder-Vardi Theorem by proving that every Monotone Monadic Strict NP problem is polynomially equivalent to a CSP problem. We showed that the class of CSP problems that can be approximated by a linear program coincides with the class of problems of width one. We proved that the equivalence problem is coNP-complete for functionally complete algebras. We showed that the sigma equivalence problem can be solved in polynomial time for commutative rings and is coNP-complete if the factor by the Jacobson radical is not commutative. We introduced the extended equivalence and equation solvability problems and we proved the corresponding dichotomy theorems for groups
Computational Complexity of SAT, XSAT and NAE-SAT for linear and mixed Horn CNF formulas
The Boolean conjunctive normal form (CNF) satisfiability problem, called SAT for short, gets as input a CNF formula and has to decide whether this formula admits a satisfying truth assignment. As is well known, the remarkable result by S. Cook in 1971 established SAT as the first and genuine complete problem for the complexity class NP. In this thesis we consider SAT for a subclass of CNF, the so called Mixed Horn formula class (MHF). A formula F in MHF consists of a 2-CNF part P and a Horn part H. We propose that MHF has a central relevance in CNF because many prominent NP-complete problems, e.g. Feedback Vertex Set, Vertex Cover, Dominating Set and Hitting Set, can easily be encoded as MHF. Furthermore, we show that SAT remains NP-complete for some interesting subclasses of MHF. We also provide algorithms for some of these subclasses solving SAT in a better running time than O(2^0.5284n) which is the best bound for MHF so far. One of these subclasses consists of formulas, where the Horn part is negative monotone and the variable graph corresponding to the positive 2-CNF part P consists of disjoint triangles only. Regarding the other subclass consisting of certain k-uniform linear mixed Horn formulas, we provide an algorithm solving SAT in time O(k^(n/k)), for k>=4. Additionally, we consider mixed Horn formulas F in MHF for which holds: H is negative monotone, c=3. We also prove the NP-completeness of XSAT for CNF formulas which are l-regular meaning that every variable occurs exactly l times, where l>=3 is a fixed integer. On that basis, we can provide the NP-completeness of XSAT for the subclass of linear and l-regular formulas. This result is transferable to the monotone case. Moreover, we provide an algorithm solving XSAT for the subclass of monotone, linear and l-regular formulas faster than the so far best algorithm from J. M. Byskov et al. for CNF-XSAT with a running time of O(2^0.2325n). Using some connections to finite projective planes, we can also show that XSAT remains NP-complete for linear and l-regular formulas that in addition are l-uniform whenever l=q+1, where q is a prime power. Thus XSAT most likely is NP-complete for the other values of l>= 3, too. Apart from that, we are interested in exact linear formulas: Here each pair of distinct clauses has exactly one variable in common. We show that NAESAT is polynomial-time decidable restricted to exact linear formulas. Reinterpreting this result enables us to give a partial answer to a long-standing open question mentioned by T. Eiter: Classify the computational complexity of the symmetrical intersecting unsatisfiability problem (SIM-UNSAT). Then we show the NP-completeness of XSAT for monotone and exact linear formulas, which we can also establish for the subclass of formulas whose clauses have length at least k, k>=3. This is somehow surprising since both SAT and not-all-equal SAT are polynomial-time solvable for exact linear formulas. However, for k=3,4,5,6 we can show that XSAT is polynomial-time solvable for the k-uniform, monotone and exact linear formula class
Graph Homomorphism, Monotone Classes and Bounded Pathwidth
A recent paper describes a framework for studying the computational
complexity of graph problems on monotone classes, that is those omitting a set
of graphs as a subgraph. If the problems lie in the framework, and many do,
then the computational complexity can be described for all monotone classes
defined by a finite set of omitted subgraphs. It is known that certain
homomorphism problems, e.g. -Colouring, do not sit in the framework. By
contrast, we show that the more general problem of Graph Homomorphism does sit
in the framework.
The original framework had examples where hard versus easy were NP-complete
versus P, or at least quadratic versus almost linear. We give the first example
of a problem in the framework such that hardness is in the polynomial hierarchy
above NP. Considering a variant of the colouring game as studied by Bodlaender,
we show that with the restriction of bounded alternation, the list version of
this problem is contained in the framework. The hard cases are
-complete and the easy cases are in P.
The cases in P comprise those classes for which the pathwidth is bounded.
Bodlaender explains that Sequential -Colouring Construction Game is in P on
classes with bounded vertex separation number, which coincides with bounded
pathwidth on unordered graphs. However, these graphs are ordered with a playing
order for the two players, which corresponds to a prefix pattern in a
quantified formula. We prove that Sequential -Colouring Construction Game is
Pspace-complete on some class of bounded pathwidth, using a celebrated result
of Atserias and Oliva.
We consider several locally constrained variants of the homomorphism problem.
Like -Colouring, none of these is in the framework. However, when we
consider the bounded-degree restrictions, we prove that each of these problems
is in our framework
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