Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$such that$uv\in A(G)$implies$f(u)f(v)\in A(H)$. If, moreover, each vertex$u \in V(G)$is associated with costs$c_i(u), i \in V(H)$, then the cost of a homomorphism$f$is$\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph$H$, the minimum cost homomorphism problem for$H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph$G$, together with costs$c_i(u)$,$u\in V(G)$,$i\in V(H)$, decide whether there exists a homomorphism of$G$to$H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes

The Ideal Membership Problem and Abelian Groups

Given polynomials $f_0,\dots, f_k$ the Ideal Membership Problem, IMP for short, asks if $f_0$ belongs to the ideal generated by $f_1,\dots, f_k$. In the search version of this problem the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications, for instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA'19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP($\Gamma$). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP($\Gamma$) where $\Gamma$ is a Boolean constraint language, while Bulatov and Rafiey [ArXiv'21] advanced these results to several cases of CSPs over finite domains. In this paper we consider IMPs arising from CSPs over `affine' constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS'21] for linear equation modulo 2, and by Bulatov and Rafiey [ArXiv'21] to systems of linear equations over $GF(p)$, $p$ prime. Here we prove that if $\Gamma$ is an affine constraint language then IMP($\Gamma$) is solvable in polynomial time assuming the input polynomial has bounded degree

Hamilton Cycles in Digraphs of Unitary Matrices

A set S ⊆ V is called an q +-set (q −-set, respectively) if S has at least two vertices and, for every u ∈ S, there exists v ∈ S, v � = u such that N + (u) ∩ N + (v) � = ∅ (N − (u) ∩ N − (v) � = ∅, respectively). A digraph D is called s-quadrangular if, for every q +-set S, we have | ∪ {N + (u) ∩ N + (v) : u � = v, u, v ∈ S} | ≥ |S | and, for every q −-set S, we have | ∪ {N − (u) ∩ N − (v) : u, v ∈ S)} ≥ |S|. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture

Mediated digraphs and quantum nonlocality

A digraph D=(V,A) is mediated if for each pair x,y of distinct vertices of D, either xyA or yxA or there is a vertex z such that both xz,yzA. For a digraph D, Δ-(D) is the maximum in-degree of a vertex in D. The nth mediation number μ(n) is the minimum of Δ-(D) over all mediated digraphs on n vertices. Mediated digraphs and μ(n) are of interest in the study of quantum nonlocality. We obtain a lower bound f(n) for μ(n) and determine infinite sequences of values of n for which μ(n)=f(n) and μ(n)>f(n), respectively. We derive upper bounds for μ(n) and prove that μ(n)=f(n)(1+o(1)). We conjecture that there is a constant c such that μ(n)f(n)+c. Methods and results of design theory and number theory are used