A Fast Algorithm for Enumerating Bipartite Perfect Matchings Takeaki Uno (National Institute of Informatics, JAPAN)

Motivations Goal: find out general techniques and properties for speed up enumeration algorithms Speeding up discrete algorithms. A lot of studies have been done. Enumeration algorithms: Algorithms have been proposed for various problems. Speeding up enumeration algorithm: Only few studies have been done. Optimization => find a solution Enumeration => find many solutions Techniques, structures, and analysis do not work well. Original techniques for enumeration algorithms are required

Enumeration Problem of Bipartite Perfect Matchings For a given bipartite graph G = (V, E), output all the perfect matchings included in G per matching Matsui, Fukuda (’93) O(|E|) I (’97) O(|V|) (amortized analysis) Minimum weighted perfect matchings Matsui, Fukuda O(|E||V|) I (’01: submitting) O(|V|) This talk: O( log |V|) ( both )

Partition the problem into two subproblems, and solve the subproblems recursively 1. Choose a vertex v 2. E 1, E 2 = Partition of edges incident to v 3. Solve two subproblems: enumeration of perfect matchings including an edge of E i for i = 1,2 Subproblem of E 1 = problem of ( V, E ＼ E 2 ) Simple Algorithm E1E1 E2E2 v

To partition the problem, subproblem of ( V, E ＼ E i ) has to include at least one perfect matching For a perfect matching M and a vertex v 1. Find a perfect matching M’ ≠ M 2. E 1, E 2 = Partition of edges incident to v such that E 1 includes an edge of M E 2 includes an edge of M’ Simple Algorithm 2 M M’ v How to find M’ ? E1E1 E2E2 # iter ≤ 2 # perfect matchings

Use a directed graph DG( G, M ) · Orient edges of M in a direction · Orient other edges in the opposite direction A perfect matching M‘ exists  a directed cycle is in DG( G, M ) M‘ is obtained by exchanging edges along the cycle Finding M’ from M M Operation of an iteration = Finding a directed cycle = O( |E|)

Consider speeding up an iteration · Really take O( |E| ) time? --- in worst case, really take · Dynamic graph approach --- maintain directed cycle Similar to transitivity update, O( |V| 2 ) · Amortized analysis --- distribute time to subproblems A graph including a perfect matching requires O( |V| 2 ) time Difficulty of Speed up Needs of · improve algorithms · improve amortized analysis

“Trimming and balancing“ generalized technique for speeding up Proposed 3 years ago ( by I ) : Isaac98 · Time complexity analysis and speeding up using the analysis, specialized for enumeration algorithms · Improved several algorithms, which had not been improved by existing techniques

Corollary of “Trimming and balancing“ For an iteration x of an (recursive type) enumeration algorithm, T(x) : computation time for x Chd(x) : Children of x T* : amortized computation time for an iteration T l : maximum computation time of bottom level iterations If for a constant α >1, for any x, α T(x) - α ≤ ∑ T(y) => T* = O ( T l ) y ∈ Chd(x) If for a constant α > 1, for any x and its child y, αT(y) + α ≥ T(x) => T* = O ( T l × log ( input size ) ) ( we use this here)

“Trimming and balancing“ Trimming and balancing improves algorithms by adding trimming phase and balancing phase to each iteration Simple algorithm · satisfies no these conditions · T l = O(|E|) For each iteration, add “trimming phase” to decrease T l → O( 1 ) add “balancing phase” to satisfy T(x)-2 ≤ 4T(y)

Trimming Phase Pass reduced graphs to subproblems Remove edges · included in all perfect matchings · included in no perfect matching → included in no directed cycle Contract 3 consecutive edges into an edge Take O(|E|) time Reduce to an equivalent problem On bottom level input graph includes one perfect matching → O(1) size graph → T l = O( 1 )

Trimming Phase 2 Lemma For a trimmed graph G = (V, E) a perfect matching M, DG(G, M) includes at least f(G) = |E| - |V| +cc(G) directed cycles (cc(G) : # of connected components of G ) In a trimmed graph G = (V, E) |E| ≥ 1.25|V| → |E| - |V| ≥ 0.2|E| → G = (V, E) includes at least 0.2|E| perfect matchings → T l = O( 1 )

Balancing Phase Set E 1, E 2 in arbitrary way If ( V, E ＼ E i ) has few edges after trimming phase, re-choose E 1, E 2 f(G) = |E| - |V| + ( # of connected components of G ) → ( V, E ＼ E i ) can includes few edges but 4 f( ( V, E ＼ E i ) ) + 2 ≥ f(G)

Re-choose edge sets If ( V, E ＼ E i ) has few edges after trimming phase then DG(G, M) includes many edges included in only directed cycles including edges of E i E1E1 E2E2 Take O(|E|) time EiEi

Analysis An iteration inputting G=( V, E ) takes O(|V|+|E|) = O( f(G) ) time On bottom level, O(|V|+|E|) = O( f(G) ) = O(1) → T l = O( 1 ) From balancing phase, 4 f( (V, E ＼ E i ) ) + 4 ≥ f(G) From Theorem, time complexity = O( 1 × log n (=log m) ) per perfect matching

Conclusion Speeded up enumeration algorithm of perfect matchings, using “trimming and balancing”