Efficient Enumeration of Induced Matchings inGraphs without Short Cycles

Kazuhiro Kurita1, Kunihiro Wasa2, Takeaki Uno2, and Hiroki Arimura1

1 IST, Hokkaido University, Sapporo, Japan, {k-kurita,arim}@ist.hokudai.ac.jp2 National Institute of Informatics, Tokyo, Japan, {wasa,uno}@nii.ac.jp

Backgrounds In this paper, we study efficient algorithms for enumerating allinduced matchings in an input graph. An induced matching in a graph G, alsocalled a risk-free marriage, is a natural extension of matchings, and defined as aset M of edges such that the vertex set of M induces M itself. Induced matchingsare an important extension of matchings, and they have potential applicationsto inexact image search or ontology matching [8].

In contrast to the case for matchings [5, 9], the maximum induced matchingproblem is computationally hard. Cameron [3], and Stockmeyer and Vazirani [10]showed that it is NP-hard to find a maximum induced matching in a graph. Itis still NP-hard for bipartite, line, and planar graphs [2–4, 8], while it is poly-nomial time solvable for the following classes: interval, chordal, weakly chordal,circular-arc, trapezoid, and co-comparability graphs [2,4,8]. However, to our bestknowledge, enumeration problems for induced matchings have not been studiedwell.

Recently, a number of efficient enumeration algorithms for all substructures ofa given graph [12] have been developed for various classes such as spanning trees,bounded-sized trees, cycles and st-paths, chordless cycles, and induced trees [1,6,7,13,14]. Most of these algorithms output solutions in polynomial amortized timeor delay in input size. Among these results, however, there are a few algorithmswith optimal complexity in amortized sense, namely constant amortized timeper output [11].

Goal of this research In this paper, we present an efficient algorithm forenumerating induced matchings in a graph. Particularly, we designed the algo-rithm so that it runs efficiently for graphs without short cycles. We show thatthe algorithm enumerates all induced matchings in constant amortized time peroutput for graphs with girth 5 or more, i.e., without no cycles of length 4 orless. Furthermore, we also show that the maximum induced matching problemis NP-complete for this class in the same manner as in [3, 10].

There are few algorithms for enumerating all matchings and induced match-ings. Recently, it is shown by Uno [11] that all matchings in a graph can be enu-merated in constant amortized time. He used an amortization technique calledPush Out to amortize the cost of each iteration to many descendants. Startingfrom a naive O(m) time enumeration algorithm based on binary partition, wedevise an improved algorithm with O(∆2) amortized complexity, and finally, aconstant amortized time algorithm by similar ideas to [11] of grouping solutionswith a similar structure.

Related work and open problems The maximum matching problem hasbeen extensively studied for long years [5,9]. Edmonds [5] presented an O(mn2)time algorithm, and Hopcroft and Karp [9] improved its complexity to O(m

√n)

time for bipartite graphs.Cameron [4] pointed out an interesting one-to-one correspondence between

induced matchings in a graph and independent sets in the square L(G)2 of theline-graph of G. Thus, constant amortized time enumeration algorithms for theformer class may give some insight to enumeration of independent sets in thelatter classes, which is still open.

In this paper, we also discuss an enumeration problem for k-distance match-ings in a graph. A k-distance matching M of a graph G is an edge set such thatfor any pair of edges e and f in M , the distance between the nearest pair ofend points of e and f is at least k. Note that a k-distance matching is a gener-alization of matchings and induced matchings. However, it seems to be difficultto develop an efficient enumeration algorithm for k-distance matchings by usingthe approach of us and Uno [11] when k ≥ 3.

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