Computing the Girth of a Planar Graph in Linear Time

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin

Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles

Plotkin, Rao, and Smith (SODA'97) showed that any graph with $m$ edges and $n$ vertices that excludes $K_h$ as a depth $O(\ell\log n)$-minor has a separator of size $O(n/\ell + \ell h^2\log n)$ and that such a separator can be found in $O(mn/\ell)$ time. A time bound of $O(m + n^{2+\epsilon}/\ell)$ for any constant $\epsilon > 0$ was later given (W., FOCS'11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has the same separator size and running time O(\mbox{poly}(h)\ell m^{1+\epsilon}). This is a significant improvement for small $h$ and $\ell$. If $\ell = \Omega(n^{\epsilon'})$ for an arbitrarily small chosen constant $\epsilon' > 0$, we get a time bound of O(\mbox{poly}(h)\ell n^{1+\epsilon}). The second algorithm achieves the same separator size (with a slightly larger polynomial dependency on $h$) and running time O(\mbox{poly}(h)(\sqrt\ell n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when $\ell = \Omega(n^{\epsilon'})$. Our third algorithm has running time O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when $\ell = \Omega(n^{\epsilon'})$. It finds a separator of size O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n) which is no worse than previous bounds when $h$ is fixed and $\ell = \tilde O(n^{1/4})$. A main tool in obtaining our results is a novel application of a decremental approximate distance oracle of Roditty and Zwick.Comment: 16 pages. Full version of the paper that appeared at ICALP'14. Minor fixes regarding the time bounds such that these bounds hold also for non-sparse graph

Inferring the Dynamics of the State Evolution During Quantum Annealing

To solve an optimization problem using a commercial quantum annealer, one has to represent the problem of interest as an Ising or a quadratic unconstrained binary optimization (QUBO) problem and submit its coefficients to the annealer, which then returns a user-specified number of low-energy solutions. It would be useful to know what happens in the quantum processor during the anneal process so that one could design better algorithms or suggest improvements to the hardware. However, existing quantum annealers are not able to directly extract such information from the processor. Hence, in this work we propose to use advanced features of D-Wave 2000Q to indirectly infer information about the dynamics of the state evolution during the anneal process. Specifically, D-Wave 2000Q allows the user to customize the anneal schedule, that is, the schedule with which the anneal fraction is changed from the start to the end of the anneal. Using this feature, we design a set of modified anneal schedules whose outputs can be used to generate information about the states of the system at user-defined time points during a standard anneal. With this process, called "slicing", we obtain approximate distributions of lowest-energy anneal solutions as the anneal time evolves. We use our technique to obtain a variety of insights into the annealer, such as the state evolution during annealing, when individual bits in an evolving solution flip during the anneal process and when they stabilize, and we introduce a technique to estimate the freeze-out point of both the system as well as of individual qubits

Initial state encoding via reverse quantum annealing and h-gain features

Quantum annealing is a specialized type of quantum computation that aims to use quantum fluctuations in order to obtain global minimum solutions of combinatorial optimization problems. D-Wave Systems, Inc., manufactures quantum annealers, which are available as cloud computing resources, and allow users to program the anneal schedules used in the annealing computation. In this paper, we are interested in improving the quality of the solutions returned by a quantum annealer by encoding an initial state. We explore two D-Wave features allowing one to encode such an initial state: the reverse annealing and the h-gain features. Reverse annealing (RA) aims to refine a known solution following an anneal path starting with a classical state representing a good solution, going backwards to a point where a transverse field is present, and then finishing the annealing process with a forward anneal. The h-gain (HG) feature allows one to put a time-dependent weighting scheme on linear ($h$) biases of the Hamiltonian, and we demonstrate that this feature likewise can be used to bias the annealing to start from an initial state. We also consider a hybrid method consisting of a backward phase resembling RA, and a forward phase using the HG initial state encoding. Importantly, we investigate the idea of iteratively applying RA and HG to a problem, with the goal of monotonically improving on an initial state that is not optimal. The HG encoding technique is evaluated on a variety of input problems including the weighted Maximum Cut problem and the weighted Maximum Clique problem, demonstrating that the HG technique is a viable alternative to RA for some problems. We also investigate how the iterative procedures perform for both RA and HG initial state encoding on random spin glasses with the native connectivity of the D-Wave Chimera and Pegasus chips.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0500

On-line and dynamic algorithms for shortest path problems

We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only $O(\log n)$ time, where $n$ is the number of vertices of the digraph. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem. Our results can be extended to hold for digraphs of genus $o(n)$