Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs Erik D. Demaine∗ MohammadTaghi Hajiaghayiy Ken-ichi Kawarabayashiz Abstract between the pieces. For example, many optimization prob- We prove two structural decomposition theorems about graphs ex- lems can be solved exactly on graphs of bounded treewidth; cluding a fixed odd minor H, and show how these theorems can what graphs can be partitioned into a small number k of be used to obtain approximation algorithms for several algorithmic bounded-treewidth pieces? In many cases, each piece gives problems in such graphs. Our decomposition results provide new a lower/upper bound on the optimal solution for the entire structural insights into odd-H-minor-free graphs, on the one hand graph, so solving the problem exactly in each piece gives a generalizing the central structural result from Graph Minor The- k-approximation to the problem. Such a decomposition into ory, and on the other hand providing an algorithmic decomposition bounded-treewidth graphs would also be practical, as many into two bounded-treewidth graphs, generalizing a similar result for NP-hard optimization problems are now solved in practice minors. As one example of how these structural results conquer dif- using dynamic programming on low-treewidth graphs; see, ficult problems, we obtain a polynomial-time 2-approximation for e.g., [Bod05, Ami01, Tho98]. Recently, this decomposi- vertex coloring in odd-H-minor-free graphs, improving on the pre- tion approach has been successfully used for graph coloring, 1−" vious O(jV (H)j)-approximation for such graphs and generalizing which is inapproximable within n for any " > 0 unless the previous 2-approximation for H-minor-free graphs. The class P = NP [Zuc07], yet has a 2-approximation in any minor- of odd-H-minor-free graphs is a vast generalization of the well- closed graph family by this approach [DHK05]. Refinements studied H-minor-free graph families and includes, for example, all of this approach have led to PTASs for many graph problems bipartite graphs plus a bounded number of apices. Odd-H-minor- [DHK05, Bak94, Epp00]. free graphs are particularly interesting from a structural graph the- Decomposition of graphs into computationally simpler ory perspective because they break away from the sparsity of H- pieces goes back to the work of Nash-Williams in 1964 minor-free graphs, permitting a quadratic number of edges. [NW64] and of Chartrand et al. [CGH71]. In particular, Chartrand et al. [CGH71] conjectured in 1971 that any 1 Introduction planar graph can have its edges partitioned into two pieces, each inducing an outerplanar graph. This conjecture was Decomposition or partitioning of graphs into smaller pieces only just proved by Goncalves [Gon05], together with a is a fundamental way to design graph algorithms. One linear-time algorithm for computing the decomposition. In of the most famous such decompositions is the divide- particular, this result establishes that any planar graph can be and-conquer separator decomposition for planar graphs of decomposed into two graphs of treewidth 2, a result proved Lipton and Tarjan [LT80], which has been generalized to earlier [Ked96]. Ding et al. [DOSV00] proved that every arbitrary graphs via sparsest cut [LR99, ARV04]. These bounded-genus graph can be decomposed into two graphs of decompositions are based on finding relatively small cuts in bounded treewidth (where the bound depends on the genus). the graph to minimize the interaction between the pieces. To Thomas [Tho95] conjectured that every graph excluding a make the pieces relatively small, the decompositions cut the fixed minor has such a decomposition. This conjecture has graph into many pieces. been proved in two papers [DDO+04, DHK05]; the latter A different kind of decomposition that has received proof is both simpler and algorithmic. A recent extension of more attention recently is to partition the graph into a this result [DHM07] allows the bounded-treewidth pieces to small number of computationally simpler (but not neces- be formed by contractions instead of deletions/subgraphs. sarily small) pieces, without much regard to the interaction In this paper, we generalize the decomposition result to odd-minor-free graphs: every graph excluding a fixed ∗MIT Computer Science and Artificial Intelligence Laboratory, 32 Vas- sar Street, Cambridge, MA 02139, USA. [email protected] odd minor can be decomposed into two bounded-treewidth yAT&T Labs — Research, 180 Park Ave., Florham Park, NJ 07932, graphs. The family of odd-H-minor-free graphs is strictly USA, [email protected]. This work was performed while the more general than H-minor-free graphs for any graph H; for author was at Carnegie Mellon University. example, it includes all complete bipartite graphs K . z n=2;n=2 National Institute for Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, In interesting contrast, the Lipton-Tarjan separator decom- Tokyo 101-8430, Japan. k [email protected] position (and thus its algorithmic consequences) cannot be ized to odd-H-minor-free graphs. Our results provide the generalized to odd-minor-free graphs, because small sep- first step in this direction by providing new algorithmic arators do not exist. Another contrast between odd-H- structural tools for odd-minor-free graphs. In particular, minor-free graphs and H-minor-free graphs is that the for- the minor-free analogs of our decomposition results have mer can have a quadratic number of edges, while the latter found many applications in just the past few years—see, e.g., are always sparse. Odd-minor-free graphs have been con- [BBCH07, APS07, ASS07, Kle05, DFHT05, DHM]—and sidered extensively in the graph theory literature (see, e.g., we analogously expect our new tools to enable several new [Gue01, GGG+04, Gue05, JT95]) and recently in theoreti- algorithmic results for odd-minor-free graphs. cal computer science [KM06]. Our immediate goal is not to develop practical algo- We prove our result by generalizing a structural result rithms: it is not clear where odd-minor-free graphs would that is the heart of Graph Minor Theory [RS03], which arise in practice, and even most algorithms for H-minor- has many algorithmic applications [Gro03, DFHT05, DH05, free graphs are currently impractical because of huge bounds DHK05], to the case of odd-minor-free graphs. Specifically, from Graph Minors. Rather, the goal of this paper is to im- we prove that every odd-minor-free graph can be decom- prove our algorithmic structural understanding, in particu- posed into a clique-sum of “almost-bipartite” graphs and lar powerful decompositions enabling efficient approxima- graphs that are “almost-embeddable” into bounded-genus tion algorithms, beyond standard graph families. surfaces. (In the original Graph Minor version, only graphs of the second type exist.) Our primary challenge in this result 1.1 Odd-Minor-Free Graphs and Their Importance. is that odd-minor-free graphs can be dense, and this density Recall that a graph H is a minor of G if H can be obtained may be equally spread throughout the graph, so we cannot by contracting and deleting edges in G. Equivalently, H is a hope to find small separations to split the graph into pieces as minor of G precisely if there are jV (H)j vertex-disjoint trees in previous decomposition theorems. Nonetheless, we show in G, one tree Tv for each vertex v of H, such that for every how to decompose odd-minor-free graphs, by showing a con- edge e = fv; wg in H there is an edge e^ in G connecting the nection to the existence of clique minors; this algorithmic two corresponding trees Tv and Tw. Now H is an odd minor technique may be useful for future structural algorithms on of G if, in addition, all the vertices of the trees can be two- dense graphs. colored in such a way that (1) the edges within each tree Tv in All of our results are algorithmic: the decompositions G are bichromatic, while (2) the edge e^ connecting trees Tv can be computed in polynomial time. Our decompositions and Tw in G corresponding to each edge e = fv; wg of H is can also be used to obtain a variety of approximation algo- monochromatic. In particular, the class of odd-H-minor-free rithms for problems on odd-minor-free graphs. In particular, graphs (excluding a fixed graph H as an odd minor) is more our decomposition into two bounded-treewidth graphs (The- general than the class of H-minor-free graphs (excluding a orem 1.1 below) immediately leads to a 2-approximation for fixed graph H as a minor). many NP-complete problems in odd-minor-free graphs. For Indeed, the class of odd-H-minor-free graphs is strictly example, our 2-approximation for graph coloring in odd-Kk- more general: the complete bipartite graph Kn=2;n=2 cer- minor-free graphs improves the recent O(k)-approximation tainly contains a Kk minor for k ≤ n=2, but on the other algorithm [KM06] and generalizes the 2-approximation for hand, it does not contain Kk as an oddp minor for k ≥ 3. In Kk-minor-free graphs [DHK05]. We also believe that com- fact, any Kk-minor-free graph G is O(k log k)-degenerate, bining our clique-sum decomposition (Theorem 1.2 below) i.e, everyp induced subgraph has a vertex of degree at most with Grohe’s technique [Gro03] leads to PTASs for maxi- O(k log k); see [Kos84,p Tho01]. Thus, any Kk-minor- mum independent set, maximum clique, and minimum ver- free graph G has O(k log kn) edges. On the other hand, tex cover. Grohe’s technique for minor-free graphs uses the some odd-Kk-minor-free graphs such as Kn=2;n=2 may have property that removing small sets of vertices in such graphs Θ(n2) edges.