Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation

Abstract

A degree-constrained graph orientation of an undirected graph G is an assignment of a direction to each edge in G such that the outdegree of every vertex in the resulting directed graph satisfies a specified lower and/or upper bound. Such graph orientations have been studied for a long time and various characterizations of their existence are known. In this paper, we consider four related optimization problems introduced in reference (Asahiro et al. LNCS 7422, 332–343 (2012)): For any fixed non-negative integer W, the problems MAX W-LIGHT, MIN W-LIGHT, MAX W-HEAVY, and MIN W-HEAVY take as input an undirected graph G and ask for an orientation of G that maximizes or minimizes the number of vertices with outdegree at most W or at least W. As shown in Asahiro et al. LNCS 7422, 332–343 (2012)).

A Appendix: An alternative proof of the submodularity of g 1

Suppose E={e ₁,e ₂,…,e _m }. Let $U = \bigcup _{i=1}^{W+1} \{v^{(i)} \,:\, v\in V\}$ be a set of W+1 copies of the vertices of V. To represent adjacency between the endpoints of e _i in G, define E _i,j ={{e _i ,u ^(j)},{e _i ,v ^(j)} : e _i ={u,v}∈E}. For each edge e _i , let E _i denote $\bigcup _{j=1}^{W+1} E_{i,j}$.

Consider a bipartite graph $H = (E\cup U, E')$ with (W+1)n+m vertices and 2(W+1)m edges, where $E' = \bigcup _{i=1}^{m} E_{i}$. Every matching M in H can be defined as a subset of E′. Then, the family of matchings is denoted by $\mathcal {M} = \{M \,:\, M \ \mbox {is a matching in}\ H\}$. Here it is easy to see that a pair $(E', \mathcal {M})$ is a transversal matroid induced by $\mathcal {E} = \{E_{1}, E_{2}, \ldots , E_{m}\}$.

We now show the correspondence between the matroid $(E', \mathcal {M})$ and the objective value of P ₁(G,W,S). The key observation is that a matching edge {e,u} in H corresponds to orienting the edge e away from the vertex u in G. We can make an orientation of G based on a matching in H: If an edge e is an endpoint of the matching edge {e,u} (called type I), then orient e away from u in G. After that, orient the remaining edges (type II) arbitrarily in G.

For each v, let C _v denote $\bigcup _{i=1}^{W+1} \{v^{(i)}\}$. Consider the induced subgraph $H' = H[E\cup \bigcup _{v\in V\setminus S} C_{v}]$ of H and a maximum matching M ^∗ in H′. For each $v\in \bigcup _{u \in V\setminus S} C_{u}$ in H′, the number n _v of vertices in C _v covered by M ^∗ is equal to $\min \{W+1, d^{+}_{\Lambda }(v)\}$ under the above constructed orientation Λ of G: Since H has only W+1 copies of each vertex u, n _v is clearly at most W+1. To obtain a contradiction, assume $d^{+}_{\Lambda }(v) < W+1$ and $n_{v} \neq d^{+}_{\Lambda }(v)$. If $n_{v} < d^{+}_{\Lambda }(v)$, there is an edge e={v,u} of type II for some u in G. The existence of such an edge in G implies that the edges of the form {e,v ⁽ⁱ⁾} and {e,u ⁽ⁱ⁾} exist in H but none of them are included in M ^∗, which contradicts the optimality of M ^∗. In addition, the above procedure to construct an orientation guarantees that $n_{v} \leq d^{+}_{\Lambda }(v)$. Hence, the optimal value of P ₁(G,W,S), i.e., $\max _{\Lambda \in \mathcal {O}(G)} {\sum }_{v\in V\setminus S} \min \{W+1, d^{+}_{\Lambda }(v)\}$ equals the size of a maximum matching in H′.

Finally, we verify the submodularity of g ₁. For $T\subseteq E'$, consider the subgraph H[T] of H. The size of a maximum matching in H[T] is equal to the rank function of the matroid $(E',\mathcal {M})$:

$$r_{\mathcal{M}}(T) = \max\{|Z| \,:\, Z\subseteq T, Z\in \mathcal{M}\}$$

For a subset $S\subseteq V$ of G, we can define $T_{S} \subseteq E'$ as {{e,v} : {e,v}∈E′,v∉S}. Hence, the optimal value of P ₁(G,W,S) can be rewritten as $r_{\mathcal {M}}(T_{S})$. It is well known that the rank function of a (transversal) matroid is submodular and that the sum of two submodular functions is submodular (e.g., [25]). It follows that g ₁(S)=r(T _S )+|S|(W+1) is submodular since |S|(W+1) is also submodular.