An official website of the United States government
We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm.
more »
« less
Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.
more »
« less
- Award ID(s):
- 1816695
- PAR ID:
- 10106371
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- ISSN:
- 1868-8969
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given n d-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training classifiers with constant margin by Clarkson et al. runs in Õ (n+d), which is also optimal in its input/output model. We design sublinear quantum algorithms for the same task running in Õ (\sqrt{n}+\sqrt{d}), a quadratic improvement in both n and d. Moreover, our algorithms use the standard quantization of the classical input and generate the same classical output, suggesting minimal overheads when used as subroutines for end-to-end applications. We also demonstrate a tight lower bound (up to poly-log factors) and discuss the possibility of implementation on near-term quantum machines.more » « less
-
null (Ed.)We consider the classical Minimum Balanced Cut problem: given a graph $$G$$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $$n$$-vertex $$m$$-edge graph $$G$$ and any parameter $$1\leq r\leq O(\log n)$$, computes a $$(\log m)^{r^2}$$-approximation for Minimum Balanced Cut on $$G$$, in time $$O\left ( m^{1+O(1/r)+o(1)}\cdot (\log m)^{O(r^2)}\right )$$. In particular, we obtain a $$(\log m)^{1/\epsilon}$$-approximation in time $$m^{1+O(1/\sqrt{\epsilon})}$$ for any constant $$\epsilon$$, and a $$(\log m)^{f(m)}$$-approximation in time $$m^{1+o(1)}$$, for any slowly growing function $$m$$. We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $$G$$ that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an $$n$$-vertex graph is $$n^{o(1)}$$, thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in $$n$$ factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs.more » « less
-
null (Ed.)Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph structured linear systems, in the general setting, the bit complexity of solving an $$n \times n$$ linear system $Ax=b$ is $$\tilde{O}(n^\omega)$$, where $$\omega < 2.372864$$ is the matrix multiplication exponent. Improving on this has been an open problem even for sparse linear systems with poly$(n)$ condition number. In this paper, we present an algorithm that solves linear systems in sparse matrices asymptotically faster than matrix multiplication for any $$\omega > 2$$. This speedup holds for any input matrix $$A$$ with $$o(n^{\omega -1}/\log(\kappa(A)))$$ non-zeros, where $$\kappa(A)$$ is the condition number of $$A$$. For poly$(n)$-conditioned matrices with $$\tilde{O}(n)$$ nonzeros, and the current value of $$\omega$$, the bit complexity of our algorithm to solve to within any $$1/\text{poly}(n)$$ error is $$O(n^{2.331645})$$. Our algorithm can be viewed as an efficient, randomized implementation of the block Krylov method via recursive low displacement rank factorizations. It is inspired by the algorithm of [Eberly et al. ISSAC `06 `07] for inverting matrices over finite fields. In our analysis of numerical stability, we develop matrix anti-concentration techniques to bound the smallest eigenvalue and the smallest gap in eigenvalues of semi-random matrices.more » « less
-
Abernethy, Jacob; Agarwal, Agarwal (Ed.)Motivated by problems in controlled experiments, we study the discrepancy of random matrices with continuous entries where the number of columns $$n$$ is much larger than the number of rows $$m$$. Our first result shows that if $$\omega(1) = m = o(n)$$, a matrix with i.i.d. standard Gaussian entries has discrepancy $$\Theta(\sqrt{n} \, 2^{-n/m})$$ with high probability. This provides sharp guarantees for Gaussian discrepancy in a regime that had not been considered before in the existing literature. Our results also apply to a more general family of random matrices with continuous i.i.d. entries, assuming that $$m = O(n/\log{n})$$. The proof is non-constructive and is an application of the second moment method. Our second result is algorithmic and applies to random matrices whose entries are i.i.d. and have a Lipschitz density. We present a randomized polynomial-time algorithm that achieves discrepancy $$e^{-\Omega(\log^2(n)/m)}$$ with high probability, provided that $$m = O(\sqrt{\log{n}})$$. In the one-dimensional case, this matches the best known algorithmic guarantees due to Karmarkar–Karp. For higher dimensions $$2 \leq m = O(\sqrt{\log{n}})$$, this establishes the first efficient algorithm achieving discrepancy smaller than $$O( \sqrt{m} )$$.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
