Independent Sets From an Algebraic Perspective

Abstract

Communicated by S. MargolisIn this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen-Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P . Moreover, we present a family of radical zero-dimensional complete intersection ideals J P associated to a finite poset P , for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by J P (that is, the number of zeros of J P ) using Gröbner methods lies in the description of the standard monomials.

Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

“…Particularly, the presence of the monomial as generator of the ideal I r , s , n involves the nonexistence of the symbol k twice in the j t h column; that of involves the nonexistence of the symbol k twice in the i t h row; and that of involves the nonexistence of 2 distinct symbols in the cell ( i , j ). On the basis of this result, the specialized algorithm described by Dickenstein and Tobis was implemented in Falcón for computing the cardinality of , for all r , s , n ≤4. For higher orders, however, the required computational cost turned out to be excessive because of large memory storage requirements.…”

“…This computational algebraic method has been implemented in the procedure PLR of the library pls.lib , available online on http://personales.us.es/raufalgan/LS/pls.lib, for the open computer algebra system for polynomial computations SINGULAR . The correctness and termination of this procedure are based on those of the algorithms described in the literature() for computing Hilbert functions. To test its efficiency, we have firstly checked the known cardinality of , for all r , s , n ≤4 (see Table ), which was already computed in Falcón .…”

Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

“…The initial ideal I r,s,n < lex which appears in the proof of Theorem 2 is the modified edge ideal [10] of the graph of rsn vertices labeled by the variables x 111 , . .…”

“…Its standard monomials can then be identified with the independent sets of such a graph. It is the fundamental of a specialized algorithm exposed by Dickenstein and Tobis [10] to compute the Hilbert series related to this kind of ideals. We have implemented this algorithm in a procedure called PLR in the open computer algebra system for polynomial computations Singular [9].…”

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

“…Particularly, the presence of the monomial x ijk x i ′ jk as generator of the ideal I r,s,n involves the non-existence of the symbol k twice in the j th column; that of x ijk x ij ′ k involves the non-existence of the symbol k twice in the i th row; and that of x ijk x ijk ′ involves the non-existence of two distinct symbols in the cell (i, j). Based on this result, the specialized algorithm described by Dickenstein and Tobis [14] was implemented in [16] for computing the cardinality of R r,s,n:m , for all r, s, n ≤ 4. For higher orders, however, the required computational cost turned out to be excessive due to large memory storage requirements.…”