970 research outputs found
Holant Problems for Regular Graphs with Complex Edge Functions
We prove a complexity dichotomy theorem for Holant Problems on 3-regular
graphs with an arbitrary complex-valued edge function. Three new techniques are
introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue
Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in
combination succeed in proving #P-hardness; and (3) algebraic symmetrization,
which significantly lowers the symbolic complexity of the proof for
computational complexity. With holographic reductions the classification
theorem also applies to problems beyond the basic model.Comment: 19 pages, 4 figures, added proofs for full versio
Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems
Constraint satisfaction problems have been studied in numerous fields with
practical and theoretical interests. In recent years, major breakthroughs have
been made in a study of counting constraint satisfaction problems (or #CSPs).
In particular, a computational complexity classification of bounded-degree
#CSPs has been discovered for all degrees except for two, where the "degree" of
an input instance is the maximal number of times that each input variable
appears in a given set of constraints. Despite the efforts of recent studies,
however, a complexity classification of degree-2 #CSPs has eluded from our
understandings. This paper challenges this open problem and gives its partial
solution by applying two novel proof techniques--T_{2}-constructibility and
parametrized symmetrization--which are specifically designed to handle
"arbitrary" constraints under randomized approximation-preserving reductions.
We partition entire constraints into four sets and we classify the
approximation complexity of all degree-2 #CSPs whose constraints are drawn from
two of the four sets into two categories: problems computable in
polynomial-time or problems that are at least as hard as #SAT. Our proof
exploits a close relationship between complex-weighted degree-2 #CSPs and
Holant problems, which are a natural generalization of complex-weighted #CSPs.Comment: A4, 10pt, 23 pages. This is a complete version of the paper that
appeared in the Proceedings of the 17th Annual International Computing and
Combinatorics Conference (COCOON 2011), Lecture Notes in Computer Science,
vol.6842, pp.122-133, Dallas, Texas, USA, August 14-16, 201
A Dichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs
We determine the computational complexity of approximately counting the total
weight of variable assignments for every complex-weighted Boolean constraint
satisfaction problem (or CSP) with any number of additional unary (i.e., arity
1) constraints, particularly, when degrees of input instances are bounded from
above by a fixed constant. All degree-1 counting CSPs are obviously solvable in
polynomial time. When the instance's degree is more than two, we present a
dichotomy theorem that classifies all counting CSPs admitting free unary
constraints into exactly two categories. This classification theorem extends,
to complex-weighted problems, an earlier result on the approximation complexity
of unweighted counting Boolean CSPs of bounded degree. The framework of the
proof of our theorem is based on a theory of signature developed from Valiant's
holographic algorithms that can efficiently solve seemingly intractable
counting CSPs. Despite the use of arbitrary complex weight, our proof of the
classification theorem is rather elementary and intuitive due to an extensive
use of a novel notion of limited T-constructibility. For the remaining degree-2
problems, in contrast, they are as hard to approximate as Holant problems,
which are a generalization of counting CSPs.Comment: A4, 10pt, 20 pages. This revised version improves its preliminary
version published under a slightly different title in the Proceedings of the
4th International Conference on Combinatorial Optimization and Applications
(COCOA 2010), Lecture Notes in Computer Science, Springer, Vol.6508 (Part I),
pp.285--299, Kailua-Kona, Hawaii, USA, December 18--20, 201
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights
Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over the Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric.
Holant problems are indeed read-twice #CSPs. Intuitively, some #CSPs that are #P-hard become tractable when restricted to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set F satisfies the condition, then F is of affine type or product type. Otherwise (a) Holant(F) is #P-hard; or (b) every function in F is a tensor product of functions of arity at most 2; or (c) F is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability
A complete dichotomy for complex-valued holant<sup>c</sup>
Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions, i.e. functions which only depend on the Hamming weight of the input. Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement. This full dichotomy for Holant^c answers a question that has been open for almost a decade.</p
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