On the Existence of General Factors in Regular Graphs

Let $G$ be a graph, and $H\colon V(G)\to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of $G$ is called an $H$-factor if the degree of any vertex $v$ in $F$ belongs to the set $H(v)$. This paper contains two results on the existence of $H$-factors in regular graphs. First, we construct an $r$-regular graph without some given $H^*$-factor. In particular, this gives a negative answer to a problem recently posed by Akbari and Kano. Second, by using Lov\'asz's characterization theorem on the existence of $(g, f)$-factors, we find a sharp condition for the existence of general $H$-factors in $\{r, r+1\}$-graphs, in terms of the maximum and minimum of $H$. The result reduces to Thomassen's theorem for the case that $H(v)$ consists of the same two consecutive integers for all vertices $v$, and to Tutte's theorem if the graph is regular in addition.Comment: 10 page

Combinatorial Nullstellensatz modulo prime powers and the Parity Argument

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA)

The Simultaneous Assignment Problem

This paper introduces the Simultaneous assignment problem. Let us given a graph with a weight and a capacity function on its edges, and a set of its subgraphs along with a degree upper bound function for each of them. We are also given a laminar system on the node set with an upper bound on the degree-sum in each member of the system. Our goal is to assign each edge a non-negative integer below its capacity such that the total weight is maximized, the degrees in each subgraph are below the associated degree upper bound, and the degree-sum bound is respected in each member of the laminar system. We identify special cases when the problem is solvable in polynomial time. One of these cases is a common generalization of the hierarchical $b$-matching problem and the laminar matchoid problem. This implies that both problems can be solved efficiently in the weighted, capacitated case even if both lower and upper bounds are present -- generalizing the previous polynomial-time algorithms. The problem is solvable for trees provided that the laminar system is empty and a natural assumption holds for the subgraphs. The general problem, however, is shown to be APX-hard in the unweighted case, which implies that the $d$-distance matching problem is APX-hard. Furthermore, we prove that the approximation guarantee of any polynomial-time algorithm must increase linearly in the number of the given subgraphs unless P=NP. We give a generic framework for deriving approximation algorithms, which can be applied to a wide range of problems. As an application to our problem, a constant-approximation algorithm is derived when the number of the subgraphs is a constant. The approximation guarantee is the same as the integrality gap of a strengthened LP-relaxation when the number of the subgraphs is small. Improved approximation algorithms are given when the degree bounds are uniform or the graph is sparse