In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N , these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any c...

For real polynomials in two indeterminates a classical polynomial harmonic decomposition (cf. (1) below) is extended from square-norm divisors to conic ones. The main result is then applied to obtain a full polynomial harmonic decomposition, and to solve a Dirichlet problem with polynomial boundary data. Harmonic functions are of utmost importance in analysis, geometry, and mathematical physics...

We consider the Multilinear set S defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form yI = ∏ i∈I xi, I ∈ I, where I denotes a family of subsets of {1, . . . , n} of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplifi...

In maintaining consistencies, such as GAC*, FDGAC* and weak EDGAC*, for global cost functions, Weighted CSP (WCSP) solvers rely on the projection and extension operations, which entail the computation of the cost functions’ minima. Tractability of this minimum computation is essential for efficient execution. Since projections/extensions modify the cost functions, an important issue is tractabl...

 ‎Let $V$ be an $n$-dimensional complex inner product space‎. ‎Suppose‎ ‎$H$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎Hrightarrow mathbb{C} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎Denote by $V_{chi}(H)$ the symmetry class‎ ‎of tensors associated with $H$ and $chi$‎. ‎Let $K(T)in‎ (V_{chi}(H))$ be the operator induced by $Tin‎ ‎text{End}(V)$‎. ‎Th...

We suggest the symmetrized Schrödinger equation and propose a general complex solution which is characterized by the imaginary units i and ǫ. This symmetrized Schrödinger equation appears some interesting features.

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: t...

Compilation into propositional languages finds a growing number of practical uses, including in constraint programming, diagnosis and machine learning (ML), among others. One concrete example is the use as classifiers, one natural question how to explain predictions made. This paper shows that for classifiers represented with some best-known languages, different kinds explanations can be comput...

We suggest the symmetrized Schrödinger equation and propose a general complex solution which is characterized by the imaginary units i and ǫ. This symmetrized Schrödinger equation appears some interesting features.