数理解析研究所講究録 1185 巻 2001 年 63-71 63

An algebraic approach to matching problems

Jalnes F. Geelen Department of Combinatorics&\mbox{\boldmath $\gamma$} Optimization University of Waterloo Waterloo, Ontario, Canada $\mathrm{N}2\mathrm{L}3\mathrm{G}1$ [email protected]

Abstract: In Tutte’s seminal paper on matching, he associates a skew-symmetric with a graph; this matrix is now known as the Tutte matrix. The rank of the Tutte matrix is exactly twice the size of a maximum matching in the graph. This formulation easily leads to an efficient randomized algorithm for matching. The Tutte matrix is also useful in obtaining a min-max theorem and an efficient deterministic algorithm. We review these results and look at similar formulations of other problems; namely, linear matroid intersection, linear

matroid parity, path matching, and $\mathrm{m}\mathrm{a}\mathrm{t}_{\mathrm{C}}\mathrm{h}\mathrm{i}_{1}$ forests.

Keywords: matching, matroid intersection, Tutte matrix, mixed matrix

1 Introduction ministic algorithms. We now progress to matching in general graphs; Two problems that play a fundamental role in for this we introduce “Pfaffians”. Let $I\mathrm{C}’$ be a combinatorial optimization are the maximum $V$ by $V$ skew-, where $V=$ cardinality matching problem and the matroid in- $\{1, \ldots, n\}$ . The of $I\mathrm{t}^{-}$ is the square tersection problem. This paper surveys matrix- of its Pfaffian. The Pfaffian has an expansion rank formulations of these and related problems. somewhat like the permutation expansion of the We begin by considering the size of a maximum determinant. Let $G(K)$ denote the graph (V, $E’$)

$E’=\{ij : I\mathrm{f}_{ij}\neq 0\}$ cardinality matching in a , which where , and let At $K$ denote is a special case of both of the aforementioned the set of perfect matchings of $G(K)$ . The Pfaf-

$I\mathrm{t}^{\vee}$ problems. fian of , denoted $Pf(I\mathrm{t}^{\vee})$ , is defined as follows: Let $G=(V, E)$ be a bipartite graph with bi- , (1) partition $(V_{r}, VC)$ , and let $(z_{e} : e\in E)$ be $\mathrm{P}\mathrm{f}(K):=\sum_{M\in u_{K}}\sigma M\prod Iuuv

Schrijver [2] for an excellent introduction.) Ma- with indeterminate entries. For example, the de- troids are a natural abstraction of linear indepen- terminant of a mixed matrix is a polynomial that dence. Basically, a matroid $M=(V,\mathcal{I})$ is deter- may have exponentially many terms. Lov\’asz [15] mined by a finite ground set $V$ and a collection $\mathcal{I}$ overcomes this problem by replacing the indeter- of independent sets, that satisfy certain axioms. minates with rational values; of course the rank

If $Q$ is a $V_{r}$ by $V_{\mathrm{c}}$ matrix, then $M(Q)$ denotes may decrease, but, fortunately, this is unlikely. If the matroid on ground set $V_{c}$ where $A\subseteq V_{c}$ is $I1^{-}$ is a matrix with indeterminate entries, then

$Q$ $A$ $I1^{\vee}$ independent if the columns of indexed by an evaluation of is $\mathrm{a}\mathrm{n}.\mathrm{v}$ matrix obtained from are linearly independent. We call $M(Q)$ a linear $K$ by replacing the indeterminates with rational matroid. Similarly, we can define a matroid on values. the rows of $Q$ ; this matroid is denoted by $M^{\mathrm{r}}(Q)$ . Theorem 1.1 Let $Q+X$ be an $n$ by $m$ mixed (We often use the following elementary fact: $X_{r}$ if $\tilde{X}$ matrix and let be an evaluation $X$ with en- is a basis of $M^{r}(Q)$ and $X_{c}$ is a basis $M(Q)$ of of tries chosen independently and at random then $Q[X_{r’ C}X]$ is nonsingular.) from $\{1, \ldots, m+n\}$ , then rank $(Q+X)=rank(Q+\tilde{X})$ Let $M_{1}$ and $M_{2}$ be matroids on a common with probability at least $\frac{1}{2}$ . ground set $V$ . The intersection problem for $M_{1}$ and $M_{2}$ is the problem of finding a maximum Theorem 1.2 Let $T$ be an $n$ by $n$ Tutte matrix,

$\mathrm{i}\mathcal{V}I_{1}$ cardinality common independent set of and $\tilde{T}$ and let be an evaluation of $T$ with entries $ch_{\mathit{0}-}$ $M_{2}$ . Now consider the intersection problem for linear matroids $M_{1}:=M(Q_{1})$ and $kI_{2}:=\mathrm{J}\sim I(Q2)$ $sen \dot{i}thenrankndependentlyandatra\tau=rank\tilde{T}w\dot{i}thprobab_{\dot{i}}nd_{\mathit{0}}mfrl\dot{i}ty\{om1,\ldots,natleaSt\frac{1\}}{2}.J$ on a common ground set $V:=\{1, \ldots , n\}$ . Let $(z_{1}, \ldots, z_{n})$ be algebraically independent com- We prove Theorems 1.1 and 1.2 in Section 2. muting indeterminates, and consider the follow- These theorems provide efficient randomized al- ing matrix: gorithms for computing the rank of a mixed ma- trix and determining the size of a maximum matching. The reader may not be comfortable . with a one in two chance of failure, but the $Z:=$ odds improve significantly with rcpcatcd trials. Among $n$ independent evaluations, one has the correct rank with probabilitv at least $1-1/2^{n}$ . Murota [18] proved that the size of a maxi- Let $T$ be the Tutte matrix of a graph $G$ , and mum common independent set of $M_{1}$ $i\downarrow ir_{2}$ and is $\tilde{T}$ let be an evaluation of $T$ . We know that rank $Z-n$ . Thus we have a matrix-rank formu- $\nu(G)\geq$ (rank $\tilde{T}$ ) $/2$ , however, knowing the evalu- lation of the linear matroid intersection problem. ation $\tilde{T}$ does not provide a more efficient method If $X$ is a $V_{r}$ by $V_{c}$ bipartite-matching matrix of finding a matching of size (rank $\tilde{T}$ ) $/2$ . Never- and $Q$ is a $V_{r}$ by $V_{c}$ matrix over the rationals, theless, Cheriyan [1] shows that random evalua- then we call $Q+X$ a mixed matrix. The matrix tions of the Tutte matrix contain a lot of infor- $Z$ , above, is a mixed matrix. Thus, the prob- mation about matching structure. lem of determining the rank of a mixed matrix contains the linear matroid intersection problem. Murota [18] studies mixed matrices extensively, Min-max theorems and shows that computing their rank is in fact Consider a $V_{r}$ by $V_{c}$ mixed nlatrix $Q+X$ . If equivalent to linear matroid intersection. In par- $Y_{r}\subseteq V_{r},$ $Y_{c}\subseteq V_{c}$ and $X[Y_{r}.Y_{c}]=0$ then ticular, we can compute the rank of $Q+X$ using Edmonds’ [5] matroid intersection algorithm. rank $Q+X\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q[Yr’ YC]+|V_{r}-Y_{r}|+|V_{c}-Y_{c}|$ .

Randomized algorithms Since $Q[Y_{r}, YC]$ is a rational matrix, we can easily evaluate the right side of tlle inequality above. The matrix-rank formulations do not immedi- Murota [17] proved that, for all appropriate ately provide efficient algorithms, as we cannot choice of $Y_{r}$ and $Y_{c}$ , this inequality is in fact at- efficiently perform basic operations on a matrix tained with equality; this generalized the result of 65

Hartfiel and Loewy [12], who characterized singu- (Here, odd $(H)$ denotes the number of compo- lar mixed matrices. nents of $H$ that have an odd number of vertices.) Let $T$ be the Tutte matrix of a graph $G$ . The Theorem 1.3 Let $Q+X$ be a $V_{r}$ by $V_{c}$ mixed matroid $M(T)$ is called the matching matroid of matrix. $Y_{r}\subseteq V_{r},$ $Y_{c}\subseteq V_{c}$ and $X[Y_{r}, Y]C=0$ If $G$ . A subset $X$ of $V$ is a matchable set of $G$ if then $G[X]$ , the subgraph of $G$ induced by $X$ , has a perfect matching. Note that, if $X$ is a match- rank $Q+X\leq rankQ[\mathrm{Y}_{f}, Y_{\mathrm{C}}]+|V_{r}-\mathrm{Y}_{r}^{r}|+|l_{c}^{r}’-Y|C$ . able set, then $X$ is independent in $M(T)$ . The

Moreover, there exists $Y_{r}^{*}\subseteq V_{r}$ and $Y_{c}^{*}\subseteq V_{c}$ such converse need not hold, since, unless $G$ is triv- that $X[Y_{r’ C}^{*}Y^{*}]=0$ and ial, $M(T)$ has independent sets of odd cardinality which cannot be matchable. However, consider a $Q+X\leq rank$ $Q[Y_{r}^{*}, \mathrm{Y}_{\mathrm{C}^{*}}]+|V_{r}-Y^{*}r|+|Vc-Y^{*}\mathrm{c}|$ rank . basis $X$ of $M(T)$ . By skew-symmetry, $X$ is also a basis of $M^{r}(T)$ . Therefore, $T[X]$ is nonsingu- For a bipartite-matching matrix $X$ , the above lar, and, hence, $X$ is a matchable set. Therefore, theorem is $\mathrm{K}\acute{\acute{\mathrm{o}}}\mathrm{n}\mathrm{i}\mathrm{g}’ \mathrm{s}$ Theorem: The rank of $X\dot{i}S$ the bases of $M(T)$ are the maximum cardinality equal to the minimum number of lines required to matchable sets of $G$ . cover all of the nonzero entries in $X$ . Let $D(G)$ denote the set of vertices $v\in V$ such We now consider a canonical version of Theo- $\nu(G-v)=\nu(G)$ and let $A(G)$ denote the rem 1.3 that appears in [9]; we require the fol- that . set of vertices in $V-D(G)$ that have a neighbour lowing definitions. Let $Q$ be a $V_{r}$ by $V_{c}$ ma- in $D(G)$ . It is $\mathrm{e}\mathrm{a}\mathrm{S}_{\backslash }\mathrm{y}$ to see that $D^{r}(T)=D(G)$ , trix. An element $i\in V_{r}$ is an avoidable row if however, we shall see that more surprising fact rank $Q[V_{r}-\{\dot{i}\}, V_{c}]=$ rank $Q$ . Thus, $i$ is an that $A^{r}(T)=A(G)$ . The following canonical ver- avoidable row if and only if $i$ is not a coloop sion of the Tutte-Berge Theorem is tantamount of $M^{r}(Q)$ . We define avoidable columns sim- to the Edmonds-Gallai Decomposition Theorem; ilarly. Now let $D^{r}(Q)$ and $D^{c}(Q)$ denote the see [16]. set of avoidable rows and columns respectively. For $i\in V_{c}-D^{c}(Q)$ , it is straightforward to Theorem 1.6 For any graph $G=(V, E)$ , see that $D^{r}(Q)\subseteq D^{r}(Q[V_{r’ C}V-\{i\}])$ . We let $A^{c}(Q)$ denote the set of elements $i\in V_{c}-D^{C}(Q)$ $2\iota \text{ノ}(c)=|V|-(odd(G-A(G))-|A(G)|)$ . such that $D^{r}(Q)=D^{r}(Q[V_{r’ \mathrm{C}}V-\{\dot{i}\}])$ , and let Moreover, $A^{r}(Q)=A^{c}(Q^{t})$ . These definitions were moti- $D(G)=D(G-A(c))$ . vated by the Dulmage-Mendelsohn decomposi- Theorems 1.4 and 1.6 shall be proved in Sec- tion of a bipartite graph [4]. tion 3.

Theorem 1.4 Let $Q+X$ be a $V_{r}$ by $V_{c}$ mixed matrix. If $Y_{r}^{*}=D^{r}(Q+T)$ and $Y_{c}^{*}=V_{c}-A_{c}$ Deterministic algorithms then $X[Y_{r}^{**}, Y]C=0$ and Edmonds’ has efficient augmenting path algo- rithms for both the matching problem [6] and the rank $Q+X\leq rank$ $Q[\mathrm{Y}_{r}^{*}, \mathrm{Y}_{C^{*}}]+|V_{r}-\mathrm{Y}_{r}’*|+|\mathrm{T}_{C}’-\gamma\iota_{c}^{r*}|$ . matroid intersection problem [5]. Therefore, the When applied to a bipartite-matching matrix, ranks of Tutte matrices and mixed matrices can the above theorem implies Dulmage and Mendel- be computed efficiently. We describe a different sohn’s decomposition theorem [4]. approach, based on evaluations. Since Tutte [20] introduced the Tutte matrix Let $Q+\overline{X}$ be an evaluation of a mixed matrix to prove his matching theorem, it should be little $Q+X$ . For an indeterminate $z$ in $X$ , we denote by

$\tilde{X}$ surprise that the Tutte matrix helps in proving $\tilde{X}^{arrow}(zarrow a)$ the evaluation of $X$ obtained from the following min-max theorem. be replacing the old value of $z$ with $a$ ; we call this perturbation. A heuristic algorithm for finding a Theorem 1.5 (Tutte-Berge Theorem) For good evaluation is to make perturbations if doing any graph $G=(\mathrm{t}^{7}’, E)$ , so increases the rank. Unfortunately, this method is not guaranteed to produce an evaluation with $2 \nu(G)=\min_{A\subseteq V}|V|-(odd(c-A\mathrm{I}-|A|)$ . the same rank as $Q+X$ ; see [9]. We overcome this 66

problem by considering a more refined ordering matrix-rank formulations for three other combi- on matrices than simply comparing rank. $\mathrm{n}\mathrm{a}\mathrm{t}_{0}\mathrm{r}\mathrm{i}\mathrm{a}\iota$ problems. Let $Q_{1}$ and $Q_{2}$ be $V_{r}$ by $V_{c}$ matrices. We write Let $T$ be the Tutte matrix of a graph $G=$ $Q_{1}\succeq Q_{2}$ if rank $Q_{1}>$ rank $Q_{2}$ , or rank $Q_{1}=$ (V, $E$ ). Given $A,$ $B\subseteq V$ , consider the problem rank $Q_{2}$ and $D^{r}(Q_{2})\subseteq D^{r}(Q_{1})$ . Similarly, we of computing the rank of $T[A, B]$ . Cunningham

$Q_{1}$ $\approx Q_{2}$ write if rank $Q_{1}$ $arrow-$ rank $Q_{2}$ and and Geelen [3], show that rank $T[A, B]$ can be $D^{r}(Q_{1})=D^{r}(Q_{2})$ . If $Q_{1}\succeq Q_{2}$ but $Q_{1}\not\simeq Q_{2}$ then computed by a slight variation on the algorithm we write $Q_{1}\succ Q_{2}$ . This gives a quasi-ordering of given by Theorem 1.8. The problem of comput- matrices; if $Q_{1}\succ Q_{2}$ then we say that $Q_{1}$ is more ing rank $T[A.B]$ has a . graphical interpretation; independent than $Q_{2}$ . see [3]. We will only collsider the special case Another heuristic algorithm for finding a good of deciding whether $T[A, B]$ is nonsingular when evaluation is to make perturbations if doing so in- $|A|=|B|$ . A perfect path-matching consists of a $\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{S}_{1}\mathrm{t}\mathrm{h}\mathrm{e}$ independence. This algorithm is guar- set of $|A-B|$ vertex disjoint paths from $A-B$ anteed to produce an evaluation with the same to $B-A$ and a perfect matching the subgraph rank as $Q+.X$ ; see [9]. induced by the vertices that are not covered by the paths. Cunningham and Geelen [3] show that $Q+\tilde{X}$ Theorem 1.7 $|/_{r}^{\mathit{7}}$ If is an evaluation of a $T[A, B]$ is nonsingular if and only if $G[A\cup B]$ has by $V_{c}$ mixed matrix $Q+X$ , then either rank $Q+$ a perfect path matching. $\tilde{X}=rankQ+X$ , or there exists an indet.erminate . $\iota$ $z$ in $X$ and $a\in\{1, \ldots, |V_{r}|+1\}$ such that $Q+$ $\tilde{X}(zarrow a)\succ Q+\tilde{X}$ . Matching forests

This theorem clearly provides a polynomial-time A mixed graph is a graph with both directed and undirected deterministic algorithm for computing the rank edges. We write $G:=(V, E, A)$ for a mixed of a mixed matrix. The algorithm is not partic- graph with vertex set V, undirected edge set $E$ , $A$ ularly efficient if implemented naively. However. and directed edge set . For $xy\in E$ we call $x$ $y$ $xy$ $xy\in A$ the are many ways to improve the running time. and heads of , and for we call $y$ a head of $xy$ . A $M$ In fact, it is possible to perform an iteration in matching forest is a subset of $E\cup A$ the same order of time that it takes to perform such that each vertex is the head of at most one $\lambda I$ , $\ovalbox{\tt\small REJECT}’I\cap A$ a matrix inversion. (This is not straightforward.) edge in and does not con- tain a directed cycle. t) The charitable reader will also note that, if the A vertex is covered by a $M$ $v$ initial evaluation is chosen at random, then it is nlatching forest if is a head of some edge $f1l$ likely to have near-optimal independence. in . Let $\mu(G)$ denote the maximum number of vertices that can be We can also compute the rank of a Tutte matrix covered bv a a matching for- est of $G$ . If $G$ $\mu(G)--2\nu(c)$ with a similar algorithm; see [8]. is undirected then . If $G$ is directed then $\mu(G)$ is the maximum size

$\tilde{T}$ of a $a$ directed forest. Matching forests were Theorem 1.8 Let be an evaluation of V $by$ in- $V$ Tutte matrix $T$ , then either rank $\tilde{T}=rank$ $T$ , troduced by Giles [11] who found an alternating

$T$ path algorithm for $\mu(G)$ or there exists an indeterminate $z$ in and $a\in$ computing . $\{1, \ldots, |V|\}$ such that $\tilde{T}(zarrow a)\succ\tilde{T}$ . Let $H=(V, A)$ be a directed graph, and let $(z_{e} : e\in A)$ be algebraically independent com- We prove Theorems 1.7 and 1.8 in Section 4. The $\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{l}$ indeterminates. Now define a $V$ by $V$ ma- $B$ proof of Theorem 1.7 in [8] is quite technical; we trix where, for $x\neq y,$ $B_{xy}=z_{\mathrm{e}}$ if $xy=e\in A$ present a simpler proof using the techniques of [9]. and $B_{xy}=0$ otherwise. The diagonal of $B$ is de-

fined so that the $\mathrm{r}\mathrm{o}\mathrm{w}-\theta \mathrm{u}\mathrm{m}\mathrm{s}$ of $B$ are $\mathrm{a}!1$ zero. We Path-matching call $B$ the branchings matrix of $H$ . Now, let $G=(\mathrm{T}^{\gamma}, E, A)$ be a mixed graph. let Above we described matrix-rank formulations for $T$ be the Tutte matrix of (V, $E$ ), and let $B$ be the matching and linear matroid intersection, and in- branchings nlatrix of (V, $A$ ). Webb [21] proved dicated why these formulations are useful. We that $\mu(G)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(T+B)$ . Using this formulation conclude the introduction by briefly describing Webb also proved a min-max theorem and ob- 67

tained an efficient deterministic algorithm. The was discovered independently by Zippel [22] and algorithm is more complicated than the algorithm Schwartz [19]. given by Theorem 1.8 for matching. In particu- lar, Webb’s algorithm may require perturbing two Lemma 2.1 Let $p(x_{1,\ldots,k}x)$ be a nonzero poly- variables at once (one for a directed edge and an- nomial of degree at most $d$ , and let $S$ be a finite other for an undirected edge), and uses a more subset of R. If $(\hat{x}_{1}, \ldots,\hat{x}_{k})$ is a random element $S^{k}$ refined notion of independence for evaluations. of , then $p(\hat{x}_{1}, \ldots,\hat{x}_{k})\neq 0$ with probability at least $1- \frac{d}{|S|}$ . Linear matroid parity Proof. The proof is by induction on $k$ . If $k=$ Consider the following problem. $1$ , then the the result follows from the fact that a nonzero single-variable polynomial of degree $d$ Matroid parity problem Given a $M$ matroid has at most $d$ roots. Suppose that $k>1$ , and $V$ on the ground set , and a partition $\Pi$ $=$ that the result holds for polynomials with fewer $(\pi_{1}, \ldots, \pi_{m})$ $V$ into pairs, of find a maximum size $k$ than variables. Collecting $p(x)$ in powers of $x_{k}$ collection $(\pi_{i_{1}}, \ldots T)\mathrm{g}i_{k}$ these pairs such that of . $\mathrm{t}$ we get $\pi_{i_{1}}\cup\cdots\cup\pi_{i_{k}}$ is independent in $M$ .

Lov\’asz showed that the matroid parity problem $p(x_{1}, \ldots, x_{k})=\sum_{i=0}^{d}pi(x1, \ldots, xk-1)x_{k}^{i}$ . is intractable (using the usual oracle based ap-

proach to matroid $\mathrm{N}\mathrm{P}$ algorithms) and -hard [16]. Since $p(x)$ is not identically zero, there exists More surprisingly, Lov\’asz showed that the ma- $j\in\{0, \ldots, d\}$ such that $pj(x1, \ldots, xk-1)$ is not troid parity problem can be solved efficiently if identically zero. Choose $j$ maximal such that $M$ is linear. $\nu_{\Pi}(M)$ Let denote the maximum $pj(x1, \ldots, xk-1)$ is not identically zero. Thus, by number of pairs in $\Pi$ whose union is independent the induction hypothesis, $p_{j}(\hat{X}_{1}, \ldots,\hat{x}_{k}-1)\neq 0$ in $M$ . with probability at least 1 – $\frac{d-j}{|S|}$ . Now, the Let $Q$ be a matrix with rows and columns in- probability that $p(\hat{x}_{1,\ldots,k}\hat{X})\neq 0$ given that dexed by $R$ and $V$ respectively, and let $\Pi$ be a $p_{j}(\hat{X}_{1}, \ldots,\hat{x}_{k}-1)\neq 0$ is at least 1 – $\frac{J^{-}}{|S|}$ . There- partition of $V$ into pairs. Now let $T$ be the Tutte fore, $p(\hat{x}_{1,\ldots,k}\hat{X})\neq 0$ with probability at least matrix of the graph with vertex set $R\cup V$ and edge set II, and let $(1- \frac{d-j}{|S|})(1-\frac{j}{|S|})$ $=$ $1- \frac{d}{|S|}+\frac{j(d-j)}{|S|^{2}}$

$R$ $V$

$\geq$ $1- \frac{d}{|S|}$ , $I\mathrm{t}^{-}:=VR$ . as required. 1

Then, $2\nu_{\Pi}(M(A))=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(T+K)-|V|$ ; see [10]. Proof of Theorem 1.1. Let $(Q+X)[A, B]$ be a More generally, if $T$ is a V by $V$ Tutte ma- maximal square nonsingular submatrix of $Q+X$ . trix and $K$ is a $V$ by $V$ rational skew-symmetric Then, $\det(Q+X)[A, B]$ is a polynomial of , then we call $T+K$ a mixed skew- at most rank $(Q+X) \leq\frac{m+n}{2}$ . Therefore, by symmetric matrix. It seems likely that some ana- Lemma 2.1, $(Q+\tilde{X})[A, B]$ is nonsingular with logue of Theorem 1.8 should hold for a mixed probability at least 1/2. Thus, rank $(Q+\tilde{X})=$ skew-symmetric matrix, but this problem re- rank $(Q+X)$ with probability at least 1/2, as mains open. Nevertheless, there is a min-max claimed. I formula for determining the rank of a mixed Similarly, by considering the Pfaffian of a skew- skew-symmetric matrix; see [10]. symmetric matrix we easily obtain Theorem 1.2.

2 Randomized algorithms 3 Min-max theorems

In this section we prove Theorems 1.1 and 1.2, In this section we prove Theorems 1.4 and 1.6, for which we require the following lemma which for which, we require the following lemma. 68

Lemma 3.1 Let $Q$ be a $V_{r}$ by $V_{c}$ matrix. let Lemma 3.3 Let $X+Q$ by a mixed matrix. If $i$ is $X_{r}=D^{r}(Q)$ , and let $X_{c}=V_{c}-A^{\mathrm{C}}(Q)$ . Then an avoidable row of $X+Q$ and $j$ is an avoidable column $X+Q$ then $X_{i,j}=0$ . I - of $Q[X_{r}, V_{\mathrm{c}}]$ $=$ $Q$ (1) rank rank ’ $randD^{c}(Q[X\gamma’ V_{\mathrm{c}}|\mathrm{t}^{\mathit{7}}\text{ノ}-\wedge\chi^{-}|rr]\mathrm{I}=$ $D^{r}(Q[x_{\Gamma}, V_{c}])=X_{r}$ , Proof of Theorem 1.4. Since deleting a row $X_{c}$ , and or a column decreases the rank of a matrix by at most one, we easily see that (2) rank $Q[X_{r’ C}X]$ $=$ rank $Q-|V_{r}-X_{r}|$ -

$|V_{c}$ – $X_{c}|$ $D^{r}(Q[XX_{C}]r’)$ $=$ $X_{r}$ , , and rank $Q+X\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q[Y_{r}, \}_{C}^{r}]+|V_{r}-Y_{r}|+|l^{\gamma}\mathrm{c}-\mathrm{Y}^{\Gamma}|c$ . $D^{c}(Q[X_{r}, X_{C}])=x_{c}$ . Now, by Lemma 3.1, Proof. Since $V_{r}-X_{r}$ is the set of coloops $M^{r}(Q)$ $B$ of , we see that, for any basis of rank $Q+X$ $\leq$ rank $(Q+X)[Y^{*}, Y^{*}rc]$ $M^{r}(Q[X_{r}, Vc])$ the set $B\cup(V_{r}-X_{r})$ is a basis $+|V_{r}-Y_{r}*|+|\nu r_{C^{-}}Y_{c}^{*}|$ , of $M^{\mathrm{r}}(Q)$ . Conversely, for any basis $B’$ of $M^{r}(Q)$ the set $B’\cap X_{r}$ is a basis $M^{r}(Q[x_{r’ c}V])$ . It fol- moreover, each row and column of $(Q+$ lows that rank $Q[X_{t’ C}V]=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q-(|V_{r}|-|_{arrow\chi_{r}^{arrow}}|)$ $X)[Y_{r’ c}^{*}Y^{*}]$ is avoidable. Thus, from Lemma 3.2, and $D^{r}(Q[Xr’ V\mathrm{C}])=X_{r}$ . $X[Y_{r’ c}^{**}Y]=0$ , as required. 1 Now, it is straightforward to see that We now consider Theorem 1.6, for which re-

$D^{\mathrm{c}}(Q[Xr’ Vc])$ $i$ contains $D^{c}(Q)$ . Now consider quire a little extra matroid theory. Let and $j$ be

$\dot{i}$ $i\in V^{c}-D^{c}(Q)$ . Since is not a coloop of distinct elements of a matroid $M$ . If neither $i$ nor

$Q[V_{r}, V_{c}-\{i\}]=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q-1$ $M(Q)$ , rank . There- $j$ is a coloop of $M$ and $r_{M}(V-\{i,j\})

$Q[X_{r}, V_{c}-\{\dot{i}\}]\geq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q-|V_{c}-X^{\vee}C|-$ fore, rank then we say that $i$ and ;/ are in series. It is well- $1=$ rank $Q[X_{r}, V_{c}]-1$ , with equality if and known, and easy, that series-pairs are transitive. only if the elements of $V_{\mathrm{c}}-X_{c}$ are coloops of A series-class of $\mathrm{J},I$ is a maximal set of elements $M^{r}(Q[V_{r’ \mathrm{C}}V-\{\dot{i}\}])$ . That is, $i\in V_{c}-D^{c}(Q)$ that, pairwise, are in series. is a coloop of $M^{r}(Q[V_{t}, Vc-\{i\}])$ if and only if $G$ $i\in A^{c}(Q)$ . This completes the proof of (1); (2) Lemma 3.4 (Gallai’s Lemma) If is a con- follows easily. I nected graph and $D(G)=V$ then $|V|$ is odd and $2\nu(G)=|V|-1$ .

$Q$ $V_{r}$ $V_{c}$ Lemma 3.2 Let be a by matrix, $i\in V_{r}$ Proof. Let $T$ be the Tutte matrix of $G$ . Since $j\in V_{c}$ $Q’$ and . Now let be a matrix obtained by $D(G)=V,$ $l\mathrm{t}l(T)$ has no coloops. Now, for any

$(i, j)$ $Q$ $\alpha\neq Q_{i,j}$ $\dot{i}$ changing the entry of to . If edge $vw$ of $G,$ $\nu(G-v-w)<\nu(G)$ . Therefore, $Q$ $j$ is an $v$ avoidable row of and is an avoidable and $u$) are in series. Now, by transitivity and $Q$ $Q’>rank$ $Q$ $G$ column of then rank . since is connected, $l\mathfrak{l}f(T)$ has a single series-

$V$ class; namely, . Thus, for any vertex $v,$ $V-\{v\}$ Proof. Let $X$ be a basis of $M^{r}(B)$ that does not is a matchable set. It follows that $|V|$ is odd and $i$ $Y$ contain and let be a basis of $l\mathrm{t}l^{\mathrm{C}}(Q)$ that $2\nu(G)=|V|-1$ , as required. 1 does not contain $j$ . Thus, $Q[X, Y]$ is a maxi- $m$ al nonsingular submatrix of $Q$ . Consequently, Proof of Theorem 1.6. Let $T$ be the Tutte $Q[X\cup\{\dot{i}\}, Y\cup\{j\}]$ is singular. Now, $\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}$ of $G$ . Recall that $D(G)=D^{r}(T)$ . Now, let $A:=A^{r}(T),$ $X_{r}:=D^{r}(T)$ , and $X_{c}:=V-$ $\det Q’[x\cup\{\dot{i}\}, Y\cup\{j\}]$ $A$ . Note that $-\lambda_{r}^{-}\subseteq \mathrm{a}\mathrm{Y}_{c}$ . By Lemma 3.1, each $=\det Q[x\cup\{i\}, Y\cup\{j\}]$ line of $T[_{-\chi-}r’ x_{c}]$ is avoidable. By mimicking the $\pm(\alpha-Qij)\det Q[x, \mathrm{Y}]$ proof of Lemma 3.2, we see that no indeterminate $=\pm(\alpha-Qij)\det Q[x, Y]$ occurs exactly once in $T[X_{r}, x_{C}]$ . Thus, by skew- $\neq 0$ . symmetry, $T[x_{r}, x_{C}-X_{r}]=0$ . That is, $A(G)\subseteq$ $A$ . Thus, rank $Q’\geq$ rank $Q’[X\cup\{i\}, Y\cup\{j\}]>$ By Lemma 3.1, we deduce that rank $T=$ rank $Q[X, Y]=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q$ , as required. 1 rank $T[V-A]+2|A|$ , and $D^{r}(\tau[V-A])=X_{r}$ . Essentially the same proof gives the following Thus, $2\nu(G)=2\nu(G-A)+2|A|$ , and $D(G-A)=$ result. $X_{r}=D(G)$ . Moreover, as $-4(G)\subseteq.4$ , there 69

are no edges in $G-A$ from $X_{r}$ to $X_{c}-arrow \mathrm{X}_{r}^{-}$ . $|V_{r}|-1$ other rows, the rank condition excludes at Then, since $D(G-A)=X_{r},$ $D(G[x_{r}])=X_{\Gamma}$ most one possible value for $a$ . $\mathrm{N}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{e}_{*\Re}$ . t,his and $G[x_{\mathrm{C}^{-}}X_{r}]$ has a perfect matching. Using leaves some choice for $a\in\{1, \ldots, |V_{r}|+1\}-\{\tilde{z}\}$ Lemma 3.4 and the fact that $D(G[X-r])=X_{r}^{-}$ , we such that $Q+\tilde{X}_{a}^{\vee}\succeq Q+\tilde{X}$ . 1 see that the components of $G[Xr]$ are all odd, and

$2\nu(c-A)=|V-A|-\mathrm{o}\mathrm{d}\mathrm{d}(G-A)$ $Q+_{\sim}\overline{\mathrm{X}}^{-}$ that . There- Lemma 4.3 Let be an evaluation of a $V_{r}$ fore, $2\nu(G)=$ ( $|V-A|$ –odd $(G-A)$ ) $+2|A|=$ by $V_{c}$ mixed matrix $Q+X,$ let $\mathrm{Y}_{r}^{r}:=D^{r}(Q+\tilde{X})$ $|V|-(\mathrm{o}\mathrm{d}\mathrm{d}(G-A)-|A|)$ . Now, it remains to and let $l_{c}’:=V_{c}-A^{c}(Q+\tilde{X})$ . If there exists prove that $A=A(G)$ . Suppose not, then there an indeterminate $z$ in $X[l_{r}’, Y_{C}]$ , then there exists $a\in A$ $X_{r}$ exists that has no neighbours in . Thus, $a\in\{1, \ldots, |V_{r}|+1\}$ such that $Q+\tilde{X}(zarrow a)\succ$ the components of $G[x_{r}]$ are all components of $Q+\tilde{X}$ . $G-(A-\{a\})$ , so odd $(G-(A-\{a\}))\geq \mathrm{o}\mathrm{d}\mathrm{d}(c-A)$ .

$\tilde{X}$ Hence, $|V|-(\mathrm{o}\mathrm{d}\mathrm{d}(G-(A-\{a\}))-|A-\{a\}|)<$ Proof. Suppose that $z$ takes the value $\tilde{z}$ in . By $|V|-(\mathrm{o}\mathrm{d}\mathrm{d}(G-A)-|A|)=\nu(G)$ . This is clearlv Lemma 4.2, there exists $a\in\{1, \ldots, |V_{c}|+1\}-\{\underline{\tilde{z}}\}$ a contradiction, and this completes the proof. 1 such that $Q+\tilde{X}(zarrow a)\succeq Q+\tilde{X}$ . Let $X_{a}$ denote $X^{\tilde{-}}(zarrow a)$ . We may assume that $Q+\tilde{X}_{a}\approx$ 4 Deterministic algorithms $Q+arrow\tilde{\lambda}’$ . By Theorem 3.1, $D^{r}((Q+\tilde{X})[Y_{r’ C}V])=$ $Y_{r}$ and $D_{c}((Q+-\tilde{K})[Y_{r’ C}V])=Y_{c}$ . Therefore, by Lemma $>$ rank In this section we prove Theorems 1.7 and 1.8; $3.2\mathrm{c}’ \mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{d}(Q+\tilde{X}_{a})[\}^{\Gamma}r’ VC]$ $(Q\backslash +\mathrm{I}$ for which we require the following lemmas. $\tilde{X})[l_{r}^{r}, V_{C}]$ ; contradicting Lemma 4.1.. Proof of Theorem 1.7. Let $\}_{r}’:=D^{r}(Q+$

$Q_{1}$ $Q_{2}$ $V_{r}$ $V_{c}$ Lemma 4.1 Let and be by matrices $\tilde{X})$ and $Y_{c}:=V_{c}-A^{c}(Q+\tilde{X}),$ alld suppose $Q_{1}\approx Q_{2}$ $X_{r}:=D^{r}(Q_{1})$ such that , and let . Then. that we cannot $\mathrm{i}\mathrm{m}\mathrm{I}$) $\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{e}$ the independence of rank $Q_{1}[X_{r}, VC]=rankQ_{2}[X_{\Gamma}, Vc]$ . $Q+\wedge\tilde{\chi}^{\vee}$ by such perturbations. Therefore, by Lemnua 4.3, $X[l’Y_{c}]\mathrm{r}’=0$ . Thus, rank $Q+X\leq$ Proof. This is an immediate corollarv of rank $Q[l_{r}^{r}, Y_{C}]+|V_{1}$. $-l_{r}^{r}|+|\iota_{/-}^{r}CY_{c}|$ . However,

Lemma 1 $Q+-\tilde{\chi}^{-}=$ 3.1. by Lenlma 3.1, rank rank $Q[Y_{r}, Y_{c}]+$

$|l^{r_{r}}-l_{r}^{r}|+|\mathrm{T}_{c}^{r}-Y_{c}|$ $\geq$ rank $Q+X$ . Thus Lemma 4.2 Let $Q+\tilde{X}$ be an evaluation of a rank $Q+arrow\tilde{\mathrm{X}}^{-}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q+X^{-}$ as required. 1 $V_{r}$ $V_{c}$ matrix $X+Q$ , and let $z$ be an by mixed Now consider Theorem 1.8. We omit the proof $in\swarrow\overline{\mathrm{t}}^{\vee}$ in $X$ that takes the value $\tilde{z}$ . indeterminate of the following Lemma, which is essentially the Then, $a\in\{1, \ldots, |V_{r}|+1\}-\{\tilde{z}\}$ there exists such same as the proof of Lemma 4.2. that $Q+\tilde{X}(zarrow a)\succeq Q+\tilde{X}$ . Lemma 4.4 Let $\tilde{T}$ be an evaluation of a $V$ by $V$ Proof. Consider any nonsingular submatrix $Q’+$ Tutte matrix $T$ , and $let\approx be$ an indeterminate in

$\tilde{X}’$ $Q+\overline{X}$ $Q’+\tilde{X}^{-\prime}(\approxarrow a)$ $\tilde{T}$ of . The determinant of $T$ that takes the value $\tilde{z}$ in . Then, there $exiSt_{S}$ is a nonzero linear function in $a,$ and, hence. has $a\in\{1, \ldots, |V|\}-\{\tilde{z}\}$ such that $\tilde{T}(zarrow a)\succeq\tilde{T}$ . exactly one root. Consequently, for any subnla- trix $Q’+\tilde{X}^{J}$ of $Q+\tilde{X}$ , we have rank $Q’+\tilde{X}’(\mathcal{Z}arrow$ We also omit the proof of the following Lemma, proof $a)\geq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}Q’+\tilde{X}’\mathrm{f}\mathrm{o}\Gamma$ is essentially the the of all but at most one choice which same as Lemma $\mathrm{o}\mathrm{f}a$ . 4.3.

$\tilde{X}_{a}$ $\tilde{X}(zarrow a)$ Let denote . Suppose that Lemma 4.5 Let $T$ be the Tutte matrix of a graph $D^{r}(Q+\swarrow\tilde{\mathrm{Y}})=\emptyset$ $Q+\tilde{X}=|V_{r}|$ $\tilde{T}$ rank . Then , so, $T$ $Y_{r}$ $G=(V_{\}}E)$ . let be an evaluation of , let $:=$ $Q+\tilde{X}_{a}\succ Q+\tilde{X}$ $Q+arrow\hat{\mathrm{Y}}_{a}\geq$ if and only if rank $D^{r}(\tilde{T})$ and let $Y_{c}:=l_{C^{-}}^{r}.4^{C}(\tilde{T})$ . If there exists $Q+\tilde{X}^{\vee}$ rank ; so the result holds. Now suppose an edge $\epsilon=xy\in E$ stlch that $x\in\}_{r}’$ and $y\in$ $Q+\tilde{X}<|V_{r}|$ $Q+\hat{X}_{a}^{-}\succeq Q+_{-\hat{\chi}^{-}}$ that rank . Then, $\mathrm{Y}_{C}^{r}-Y_{r}$ , then there exists $a\in\{1, ..., |V|\}$ such $(Q+\overline{X}_{a})[\mathrm{f}^{\prime_{r}}-\{i\}, l_{c}^{r}]$ if and only if rank $\geq$ that $\dot{T}(z_{\mathrm{e}}arrow a)\succ\tilde{T}$ . rank $(Q+arrow\tilde{\mathrm{Y}})[V_{r}-\{i\}, \mathrm{f}_{c}^{\Gamma}\text{ノ}]$ for all $\dot{i}\in V_{r}$ . If the in-

$(Q+-\tilde{\mathrm{Y}}^{-}a)[\mathrm{T}_{\Gamma^{-}}^{r}$ $T$ determinate $z$ is in row $j$ of $X$ then Lemma 4.6 Let be the Tutte matrix of a graph

$\tilde{T}$ $\{j\},$ $V_{c}]=(Q+\tilde{X})[V_{r}-\{j\}, V_{c}]$ ; for each of the $G=(V, E)$ . let be an evaluation of $T$ , and let 70

$Y_{r}:=D^{r}(\tilde{T})$ . If there $ex\dot{i}St_{S}$ an edge $\epsilon=xy\in E$ 2. 1V.J. Cook. W.H. Cunningham, W.R. Pul- $y\in Y_{r}$ such that $x,$ and $x$ and $y$ are in different leyblank, and A. Schrijver, Combinatorial series classes of $M(\tilde{T}[Y_{r}, V])$ then there exists $a\in$ Optimization, Wiley, New York, 1998. $\{1, \ldots, |V|\}$ such that $\tilde{T}(z_{\mathrm{e}}arrow a)\succ\tilde{T}$ . 3. W.H. Cunningham and J.F. Geelen, “Com- Proof. Let $Y_{c}:=V_{c}-A^{C}(\tilde{T})$ , and suppose that binatorial algorithms for path-matching”, in

$\tilde{T}$ $z_{e}$ takes the value $\tilde{z}_{\mathrm{e}}$ in . By Lemma 4.4, preparation. there exists $a\in\{1, \ldots, |V|\}-\{\tilde{z}_{\mathrm{e}}\}$ such that

$\mathrm{A}.\mathrm{L}$ $\mathrm{N}.\mathrm{S}$ $\tilde{T}(z_{e}arrow a)\succeq\tilde{T}$ $\tilde{T}_{a}$ 4. . and . Let denote $\tilde{T}(zarrow a)$ . Dulmage . Mendelsohn, “A

$\tilde{T}$ $\tilde{T}_{a}$ $\approx$ structural theory of $\mathrm{b}\mathrm{i}\mathrm{p},\mathrm{a}$ rtite graphs for fi- We may assume that . Note that ,

$Y_{r}$ $Y_{r}$ $\subseteq$ dimension, $Y_{c}$ llite exterior $x,$ $y\in$ and . By Theorem 3.1, Trans. Roy. Soc. Canada, $D^{r}(\tilde{T}[Yr’ V])=Y_{r}$ $x$ Section 3, vol. 53 (1959), 1-13. and, since and $y$ are not $M(\tilde{T}[Y_{r}, V]),$ in series in $x$ is an avoidable col- 5. J. Edmonds, “Submodular functions, umn of $\tilde{T}[l_{\Gamma}^{r}, V-\{y\}]$ . Therefore, by Lemma 3.2. ma- troids and certain polyhedra”, in rank $\tilde{T}_{a}[Y_{r}, V-\{y\}]>\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\tilde{T}Yr’ V-\{y\}]$ . How- Combinato- rial Structures and their Applications $(\mathrm{R}.\mathrm{K}$ . ever, rank $\tilde{T}[Y_{r}, V-\{y\}]=$ rank $\tilde{T}[Y_{r}, l\text{ノ}]r$ , so Guy et al. Eds.), $\tilde{T}[\mathrm{Y}_{r}, V_{C}]$ Gordon rank $\tilde{T}_{a}[Y_{r}, V_{c}]>$ rank ; contradicting land Breach, New York, 1970. Lemma 4.1. 1

Proof of Theorem 1.8. Let $A:=A^{c}(\tilde{T}),$ $l’,$. $:=$ 6. J. Edmonds, “Paths, trees, and flowers”, $D^{r}(\tilde{T})$ and $Y_{c}:=V-A$ ; and suppose that we can- Canad. J. Math. 17 (1965), 449-467.

$\tilde{T}$

$\mathrm{e}\Gamma-$ not improve the independence of by such $\mathrm{I}$)

$\mathrm{C}.\mathrm{D}$ turbations. Therefore, by Lemma 4.5, $T[l_{r’ c}’l’-$ 7. . Godsil, Algebraic Combinatorics,

$Y_{r}]=0$ . Thus rank $\tilde{T}[\mathrm{Y}_{r}, Y_{c}]=$ rank $\tilde{T}[l_{\Gamma}^{r}]$ . Chapman and Hall. 1993.

Moreover, by Lemma 4.6, for each edge $xy$ of $\mathrm{J}.\mathrm{F}$ $G[Y_{r}],$ 8. . Geelen, “An algebraic matching algo- $x$ and $y$ are in series in $M(\tilde{T}[Y_{r}, V])$ . The $\mathrm{C}^{\mathrm{t}}\mathrm{o}\mathrm{n}1\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a},$ $20$ $A$ rithm”, (2000), elements of are coloops of $M(\tilde{T}[Y_{\Gamma}, V])$ and 61-70. the elements in $Y_{c}-Y_{r}$ are loops of $\mathbb{J}t(\check{T}Yr’\eta)$ . 9. $\mathrm{J}.\mathrm{F}$ . Geelen. “Maximunl rank matrix com- $G[Y_{r}],$ Therefore, for each edge $xy$ of $X$ and $y$ are pletion”, Linear Algebra Appl. 288 (1999), in series in $M(\tilde{T}[Yr])$ . Hence, by the transitivity 211-217. of series-pairs, the series-classes of $M(\tilde{T}[Y_{r}])$ are

$G[Y_{r}]$ determined by the components of $.\mathrm{I}.\mathrm{F}$ . Conse- 10. . Geelen and S. Iwata, “Matroid match- quently, rank $\tilde{T}[Y_{r}]=|Y_{r}|$ $(G[l_{r}^{r}])$ -odd . Now. by ing and the rank of a mixed skew-symmetric 3.1, Lemma matrix”, in preparation. rank $\tilde{T}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\tilde{\tau}[Y_{r’ C}Y]+|V-Y_{r}|+|V-Y_{C}|$ 11. R. Giles, “Optimum lnatching forests. I. $=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\tilde{T}[Y]r+|V-Y_{r}|+|A|$ Special weights”, Math. Programming, 22, $=(|Y_{r}|$ –odd $(G[Y_{r}])+|\mathrm{T}’/-1_{r}^{r}|+|A|$ (1982) 1-11.

.. $\geq$ ( $|Y_{r}|$ –odd $(G-A)$ ) $+|V-Y_{r}|+|.4|$

$\mathrm{D}.\mathrm{J}$ $=|V|-(\mathrm{o}\mathrm{d}\mathrm{d}(G-A\mathrm{I}-|A|)$ 12. . Hartfiel and R. Loewy, “A determinan-

$\geq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}T$ . tal version of the Frobenius-K\’onig theorem”, Linear Multilinear Algebra 16, (1984), 155- Hence, rank $\tilde{T}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}T$ , as required. 1 165.

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