Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation

Yuichi Asahiro; Jesper Jansson; Eiji Miyano; Hirotaka Ono

doi:10.1007/s00224-014-9565-5

Theory of Computing Systems

January 2016, Volume 58, Issue 1, pp 60–93 | Cite as

Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation

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Authors and affiliations

Yuichi AsahiroEmail author
Jesper Jansson
Eiji Miyano
Hirotaka Ono

Article

First Online: 19 December 2014

Received: 03 February 2014

Accepted: 23 August 2014

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1 Citations

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)).

Keywords

Graph orientation Degree constraint (In)approximability Submodular function Greedy algorithm

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Notes

Acknowledgments

This work was supported by KAKENHI grant numbers 23500020, 25104521, 25330018, 26330017, and 26540005 and The Hakubi Project at Kyoto University. The authors would like to thank the anonymous reviewers for their detailed comments and suggestions that helped to improve the presentation of the paper, and Peter Floderus for pointing out an error in one of the figures.

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.

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Authors and Affiliations

Yuichi Asahiro
- 1
Email author
Jesper Jansson
- 2
Eiji Miyano
- 3
Hirotaka Ono
- 4

1.Department of Information ScienceKyushu Sangyo UniversityFukuokaJapan
2.Laboratory of Mathematical Bioinformatics, Institute for Chemical ResearchKyoto UniversityKyotoJapan
3.Department of Systems Design and InformaticsKyushu Institute of TechnologyFukuokaJapan
4.Department of Economic EngineeringKyushu UniversityFukuokaJapan

Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation

Abstract

Keywords

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Acknowledgments

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