A Real Polynomial for Bipartite Graph Minimum Weight Perfect Matchings
aa r X i v : . [ c s . D M ] M a r A Real Polynomial for Bipartite GraphMinimum Weight Perfect Matchings ∗ Thorben Tr¨obst and Vijay V. Vazirani [email protected], [email protected]
Department of Computer Science, University of California, IrvineMarch 26, 2020
Abstract
In a recent paper, Beniamini and Nisan [BN20] gave a closed-form formula for the uniquemultilinear polynomial for the Boolean function determining whether a given bipartite graph G ⊆ K n,n has a perfect matching, together with an efficient algorithm for computing thecoefficients of the monomials of this polynomial. We give the following generalization: Givenan arbitrary non-negative weight function w on the edges of K n,n , consider its set of minimumweight perfect matchings. We give the real multilinear polynomial for the Boolean functionwhich determines if a graph G ⊆ K n,n contains one of these minimum weight perfect matchings. Every Boolean function f : { , } n → { , } can be represented in a unique way as a real multilin-ear polynomial, and this and related representations have found numerous applications, e.g. see[O’D14]. In what appears to be a ground-breaking paper, Beniamini and Nisan [BN20] gave thispolynomial for the Boolean function determining whether a given bipartite graph G ⊆ K n,n has aperfect matching; importantly, they show how to efficiently compute the coefficient of any specifiedmonomial of this polynomial. They also gave a number of applications of this fact, in particular,to communication complexity.Given an arbitrary non-negative weight function w on the edges of K n,n , consider its set ofminimum weight perfect matchings. Building on the work of [BN20], we give the real multilinearpolynomial for the Boolean function which determines if a graph G ⊆ K n,n contains one of theseminimum weight perfect matchings. As above, we can also efficiently compute the coefficients ofthe monomials of this polynomial. As mentioned in Remark 18, this is a generalization of the maintheorem of [BN20]. In this section, we will give some key definitions and facts, culminating in the main theorem of[BN20]. These notions pertain to bipartite graphs only. ∗ Supported in part by NSF grant CCF-1815901. otation 1. Given a graph G , V ( G ) and E ( G ) will denote its set of vertices and edges, respectively.If G, H are two graphs, then H is a subgraph of G , denoted H ⊆ G , if V ( H ) = V ( G ) and E ( H ) ⊆ E ( G ). K n,n denotes the complete balanced bipartite graph on 2 n vertices. If S is a set of subgraphsof K n,n , then by E ( S ) we mean the union of the sets of edges of the graphs in S , i.e., E ( S ) = [ G ∈ S E ( G ) . Definition 2.
Let F be a family of subgraphs of K n,n and G be a subgraph of K n,n . We will saythat G is F -covered if ∃ S ⊆ F s.t. E ( G ) = E ( S ). The membership function for family F is a 0/1function f F on subgraphs of K n,n satisfying f F ( G ) = 1 iff ∃ H ∈ F s.t. H ⊆ G .Graph G ⊆ K n,n will be represented by the following setting of the n variables ( x , . . . , x n,n ): x i,j = 1 if ( i, j ) ∈ E ( G ) and x i,j = 0 if ( i, j ) / ∈ E ( G ). Hence we will view the membership functionfor family F as f F : { , } n → { , } . Definition 3.
Let graph G ⊆ K n,n . By the multilinear monomial corresponding to G , denoted m G ( x , , . . . , x n,n ), we mean Y ( i,j ) ∈ E ( G ) x i,j . A following important fact was proven in [BN20]:
Fact 4. ([BN20]) Let F be a family of subgraphs of K n,n . The only monomials appearing inthe multilinear real polynomial corresponding to the membership function for family F are thosecorresponding to F -covered subgraphs of K n,n . Let F be a family of subgraphs of K n,n . Let C ( F ) denote the set of all F -covered subgraphs of K n,n and ˆ0 denote the empty subgraph of K n,n . The set C ( F ) ∪ { ˆ0 } under the relation ⊆ defines aposet. Fact 5. [BN20] The poset (( C ( F ) ∪ { ˆ0 } ) , ⊆ ) is a lattice. We will denote the lattice corresponding to family F by L ( F ). Next we define a very speciallattice. Let P n,n denote the family of perfect matchings of K n,n . The P n,n -covered graphs arecalled matching covered graphs , see [LP86]. The lattice L ( P n,n ) has special properties, as shown byBillera and Sarangarajan [BS94]: Fact 6. [BS94] The lattice L ( P n,n ) is isomorphic to the face lattice of the Birkhoff polytope for K n,n . It is well known that the face lattice of the Birkhoff polytope is not only graded but also Eulerian,see [Sta96]. A lattice is graded if it admits a rank function ρ satisfying:1. If x < y then ρ ( x ) < ρ ( y ).2. If y covers x (i.e., x < y and there is no element z in the poset such that x < z < y ), then ρ ( y ) = ρ ( x ) + 1.Given any two elements x < y in the lattice, by the interval of x and y we mean the set { z | x ≤ z ≤ y } . A graded lattice is Eulerian if in every interval, the number of even-ranked and odd-rankedelements must be equal. Since L ( P n,n ) is isomorphic to the face lattice of the Birkhoff polytope, itis also Eulerian. 2 efinition 7 (see [Sta96]) . Let P = ( P, < ) be a finite partial order. Then the M¨obius function on P , µ : P × P → R , is defined as follows: ∀ x ∈ P : µ ( x, x ) = 1 ∀ x, y ∈ P, with y < x, µ ( y, x ) = − X y ≤ z M¨obius number of x .Since L ( P n,n ) is an Eulerian lattice, one can show that the M¨obius function on this lattice canbe expressed in terms of the rank function as follows: Fact 8. (see [Sta96]) For any graph G ∈ L ( P n,n ) , µ (ˆ0 , G ) = ( − ρ ( G ) . Furthermore, [BN20] show that ρ ( G ) is easy to compute using the notion of cyclomatic numberof a graph. Definition 9. For any subgraph G of K n,n , define its cyclomatic number to be χ ( G ) = E ( G ) − V ( G ) + C ( G ) , where C ( G ) is the number of connected components of G .The proof of Fact 10 uses the notion of ear-decomposition of a connected, matching-coveredbipartite graph; see [LP86] for a detailed discussion of this notion. Fact 10. [BN20] For any matching covered subgraph G of K n,n , ρ ( G ) = χ ( G ) + 1 . The last ingredient needed is: Fact 11. [BN20] Let F be a family of subgraphs of K n,n . Then the membership function for thisfamily is given by: f F ( x , , . . . , x n,n ) = X G ∈C ( F ) − µ (ˆ0 , G ) · m G ( x , , . . . , x n,n )By Fact 8, we get the M¨obius function in terms of the rank function and by Fact 10 we get therank function in terms of χ ( G ). Substituting this in Fact 11, we get: Theorem 12. [BN20] f P n,n ( x , , . . . , x n,n ) = X G ∈C ( P n,n ) ( − χ ( G ) · m G ( x , , . . . , x n,n ) Corollary 13. [BN20] The number of matching-covered subgraphs of K n,n is odd.Proof. Since L ( P n,n ) is Eulerian, the cardinality of every closed interval is even. Furthermore, sincethe closed interval [ˆ0 , K n,n ] contains all matching-covered subgraphs of K n,n and the empty graph,the lemma follows. Remark 14. Fact 11 gives the multilinear real polynomial in closed form for an arbitrary mem-bership function; however, it is in terms of the M¨obius function. It turns out that the latter isnot easy to compute in general, as we will see in Section 4. The key step taken by [BN20] wasto show how to efficiently compute the M¨obius number corresponding to any specified monomialof formula for the membership function of perfect matchings. This critically depends on the factthat the lattice P n,n is Eulerian, which follows from Fact 6, and Fact 8, which relates the rank ofa matching covered graph to its cyclomatic number. Of course the entire formula will in generalhave exponentially many monomials. 3 Extension to Minimum Weight Perfect Matchings We next turn to minimum weight perfect matchings. Let w : E ( K n,n ) → Q + be a fixed but arbitraryweight function. Definition 15. Let P wn,n denote the family of minimum weight perfect matchings of K n,n withrespect to weight function w , and f n,w : { , } n → { , } denote the membership function for thisfamily. Let C ( P wn,n ) denote the set of all P wn,n -covered subgraphs of K n,n .By Fact 5, the poset (( C ( P wn,n ) ∪ { ˆ0 } ) , ⊆ ) is a lattice, which we will denote by L ( P wn,n ). Lemma 16. Let G w be the graph whose edge set is the union of all matchings in P wn,n . Then L ( P wn,n ) ⊆ { G ∈ L ( P n,n ) | G ≤ G w } , i.e. L ( P wn,n ) is the closed interval [0 , G w ] in L ( P n,n ) .Proof. This follows from the fact that if M ⊆ G w is any perfect matching, then M ∈ P wn,n . To seethis note that the face of the optimum solutions of the LPmin w ( x )s . t . x ( δ ( a )) = 1 ∀ a ∈ A,x ( δ ( b )) = 1 ∀ b ∈ B,x e ≥ ∀ e ∈ E ( K n,n )is given by the polytope Q := x ∈ R E ( K n,n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ( δ ( a )) = 1 ∀ a ∈ A,x ( δ ( b )) = 1 ∀ b ∈ B,x e = 0 ∀ e / ∈ E ( G w ) ,x e ≥ ∀ e ∈ E. But any perfect matching M ⊆ G w defines a point in Q which means that w ( M ) is minimum. Thus M ∈ P wn,n . Theorem 17. f n,w ( x , , . . . , x n,n ) = X G ∈C ( P wn,n ) ( − χ ( G ) · m G ( x , , . . . , x n,n ) Proof. By Fact 11 we have f n,w ( x ) = X G ∈C ( P wn,n ) − µ (ˆ0 , G ) · m G ( x ) . But since L ( P wn,n ) is just the downward hull of G w in L ( P n,n ), we know that µ L ( P wn,n ) = µ L ( P n,n ) .Thus f n,w ( x ) = X G ∈C ( P wn,n ) − µ L ( P n,n ) (ˆ0 , G ) · m G ( x )= X G ∈C ( P wn,n ) ( − χ ( G ) · m G ( x ) . emark 18. As in [BN20], the coefficients of the monomials of this polynomial can also be ef-ficiently computed. Analogous to Corollary 13, the number of P wn,n -covered subgraphs of K n,n isodd; this follows for the fact that L ( P n,n ) is Eulerian. Finally, observe that Theorem 17 is a gen-eralization of Theorem 12 since for the special case of unit weights on all edges of K n,n , all perfectmatchings have the same weight. It is easy to check that most of the machinery of [BN20] extends readily to non-bipartite graphs.However, their main theorem does not extend in any straightforward manner. Let P n denote theset of perfect matchings of K n and L ( P n ) denote the lattice of matching covered subgraphs of K n .This lattice is not Eulerian, and not even graded. Therefore new structural properties are neededfor obtaining an efficient algorithm for computing the associated M¨obius numbers. The possibilitythat this function is NP-hard to compute is also not ruled out; however, it is important to pointout that over the years, for numerous algorithmic results, the non-bipartite case has followed thebipartite case, with the infusion of appropriate structural facts, which are typically quite non-trivial;see Section 1.3 in [AV20] for an extensive discussion of this phenomenon.More generally, it will be interesting to study the complexity of computing individual M¨obiusnumbers of other lattices which have polynomial representations. An important candidate is thelattice of stable matchings for which a succinct representation follows from Birkhoff’s representationtheorem [Bir37] and the notion of rotations, see [GI89].We note that the smallest value of n for which L ( P n ) is not Eulerian is n = 6. In particular,for the graph G (see Figure 1), the interval [0 , G ] (see Figure 2) is not Eulerian. Note also that µ (0 , G ) = 0 which means that the monomial m G ( x ) does not appear in the associated graph coveringpolynomial. Indeed, one can show that L ( P ) is not even graded. We exhibit an explicit pentagonsublattice in Figure 3. Figure 1: The graph G ⊆ K referred to in Figure 2.Besides non-bipartite perfect matching, one can seek real polynomials for several other graphproperties, including connected subgraphs; out-branchings from a given root vertex in a directedgraph; and subgraphs containing an s - t path, for given vertices s and t . References [AV20] Nima Anari and Vijay V Vazirani. Matching is as easy as the decision problem, in theNC model. In .5 igure 2: Interval [0 , G ] ⊆ L ( P ) for the graph of Figure 1, showing that L ( P ) is not Eulerian.Schloss Dagstuhl-Leibniz-Zentrum f¨ur Informatik, 2020.[Bir37] Garrett Birkhoff. Rings of sets. Duke Mathematical Journal , 3(3):443–454, 1937.[BN20] Gal Beniamini and Noam Nisan. Bipartite perfect matching as a real polynomial. arXivpreprint arXiv:2001.07642 , 2020.[BS94] Louis J Billera and Aravamuthan Sarangarajan. The combinatorics of permutation poly-topes. In Formal power series and algebraic combinatorics , volume 24, pages 1–23, 1994.[GI89] Dan Gusfield and Robert W Irving. The stable marriage problem: structure and algorithms .MIT press, 1989.[LP86] L. Lov´asz and M.D. Plummer. Matching Theory . North-Holland, Amsterdam–New York,1986.[O’D14] Ryan O’Donnell. Analysis of Boolean functions . Cambridge University Press, 2014.[Sta96] Richard Stanley. Enumerative Combinatorics, vol. 1 . Wadsworth and Brooks/Cole, PacificGrove, CA, 1996. 6 igure 3: A pentagon sublattice in the lattice L ( P6