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Book Synopsis Advances in the Computational Complexity of Holant Problems by :
Download or read book Advances in the Computational Complexity of Holant Problems written by and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the computational complexity of counting problems defined over graphs. Complexity dichotomies are proved for various sets of problems, which classify the complexity of each problem in the set as either computable in polynomial time or \#P-hard. These problems are expressible in the frameworks of counting graph homomorphisms, counting constraint satisfaction problems, or Holant problems. However, the proofs are always expressed within the framework of Holant problems, which contains the other two frameworks as special cases. Holographic transformations are naturally expressed using this framework. They represent proofs that two different-looking problems are actually the same. We use them to prove both hardness and tractability. Moreover, the tractable cases are often stated using a holographic transformation. The uniting theme in the proofs of every dichotomy is the technical advances achieved in order to prove the hardness. Specifically, polynomial interpolation appears prominently and is indispensable. We repeatedly strengthen and extend this technique and are rewarded with dichotomies for larger and larger classes of problems. We now have a thorough understanding of its power as well as its ultimate limitations. However, fundamental questions remain since polynomial interpolation is intimately connected with integer solutions of algebraic curves and determinations of Galois groups, subjects that remain active areas of research in pure mathematics. Our motivation for this work is to understand the limits of efficient computation. Without settling the P versus #P question, the best hope is to achieve such complexity classifications.