\( % groupings of objects. \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\seq}[1]{\left(#1\right)} \newcommand{\ang}[1]{\langle#1\rangle} \newcommand{\tuple}[1]{\left(#1\right)} \newcommand{\size}[1]{\left| #1\right|} \newcommand{\comp}{\circ} % numerical shortcuts. \newcommand{\abs}[1]{\left| #1\right|} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} % linear algebra shortcuts. \newcommand{\change}{\Delta} \newcommand{\norm}[1]{\left\| #1\right\|} \newcommand{\dprod}[1]{\langle#1\rangle} \newcommand{\linspan}[1]{\langle#1\rangle} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\der}[1]{\frac{d#1}{dx}} \newcommand{\lap}{\Delta} \newcommand{\kron}{\otimes} \newcommand{\nperp}{\nvdash} \newcommand{\mat}[1]{\left[ \begin{smallmatrix}#1 \end{smallmatrix} \right]} % derivatives and limits \newcommand{\partder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\partdern}[3]{\frac{\partial^{#3 #1}}{\partial #2^{#3}}} \newcommand{\gradient}{\nabla} \newcommand{\subdifferential}{\partial} % Arrows \newcommand{\diverge}{\nearrow} \newcommand{\notto}{\nrightarrow} \newcommand{\up}{\uparrow} \newcommand{\down}{\downarrow} % gets and gives are defined! 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Applications

For number theory applications, see number theory ref.

Perfect matchings

Bipartite graph

G = (U, V, E); \(|U| = |V| = n\). Matching: \(\set{e} \in E\) not sharing endpoints. Perfect matching (pm) has size n. Naive alg takes \(O((n!)n)\) time.

Make symbolic matrix with \(A_{i,j} = x_{i,j}\) if \((u_{i}, v_{j})\in E\), else 0. Let Q(x) = det(A) : \(n^{2}\)-nomial, ‘symbolic determinant’. G has pm iff \(\exists r: Q(r) \neq 0\). \(deg(Q) \leq n\). Take prime \(p > 2n\), pick r from \(Z_{p}\); by Schwartz Zippel Thm, \(Pr(Q(r) = 0 | \exists r’|Q(r’) \neq 0) < 2^{-1}\).

If pm unique, find pm in polytime: Repeat: \(E = E - \set{e}\); see if G still has pm. (Parallelizable.)

If pm not unique

Take uniformly random weight fn \(w:[1,m] \to [1,2m]\); let \(W(S) = \sum_{d \in S} w(d)\).

Isolation lemma

Let \(S_{i} \in [1,m]\); take k \(S_{i}\). \(Pr(\exists minWt\ S_{i}, S_{j} | e \in S_{i} , e \notin S_{j}) \leq (2m)^{-1}\): by principle of deferred decisions: Pick W(e) last to \(W(S_{j}) - W(S_{i} - \set{e})\). So, \(\union\) bound: Pr(\(\exists\) unique \(S_{i}\) with \(W(S_{i})\) min) \(\leq 2^{-1}\).

Get w(e) \(\forall e = (u_{i}, v_{j}) \in E\). Get matrix \(B_{i,j} = 2^{w((u_{i}, v_{j}))}\). Then \(det(B) = \sum_{pm\ M} (\pm 2^{W(M)})\). So find pm by finding highest power of 2 dividign det(B).

General graph

(Tutte): G = (V, E); Make Tutte matrix: \(\forall i<j\) if \((u_{i}, v_{j})\in E\), \(A_{i,j} = x_{i,j}\) and \(A_{j,i} = -x_{i,j}\), else 0. Let \(|A| = Q(x)\) multinomial; G has pm iff \(\exists x Q(x) \neq 0\): \(det(A) = \sum_{\pi} sgn(\pi) \prod A_{i, \pi(i)}\); if some \(\forall \pi: \prod A_{i, \pi(i)} = 0 \), no pm. Else, Take prime \(p > 2n\), pick r from \(Z_{p}\), take Q over \(Z_{p}\); by Schwartz Zippel, \(\exists x\) where \(Q(x) \neq 0\).