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This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions to other polytopes, discussion of other available methods and experimental results.
Recently, it was found that a new set of simple techniques allow one to conveniently express ordinary integrals through differentiation. These techniques add to the general toolbox for integration and integral transforms such as the Fourier and Lapla
In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Un
To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X({ n_x : x in X })$. This is obtained from the blowup $X[{bf n}]$ - which
In this paper, we present a new methodology to evaluate whether a business process model is fully compliant with a regulatory framework composed of a set of conditional obligations. The methodology is based failure delta-constraints that are evaluate
In this paper, we examine how far a polynomial in $mathbb{F}_2[x]$ can be from a squarefree polynomial. For any $epsilon>0$, we prove that for any polynomial $f(x)inmathbb{F}_2[x]$ with degree $n$, there exists a squarefree polynomial $g(x)inmathbb{F