Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar.
As mentioned before, the matching reconfiguration has been studied by several authors @cite_3 @cite_18 @cite_13 @cite_10 @cite_8 . Extension to @math -matchings has been considered, too @cite_2 @cite_0 .
@cite_13 @cite_18 @cite_8 @cite_3 @cite_0 @cite_2 @cite_10
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Consider a graph such that each vertex has a nonnegative integer capacity and each edge has a positive integer weight. Then, a b-matching in the graph is a multi-set of edges (represented by an integer vector on edges) such that the total number of edges incident to each vertex is at most the capacity of the vertex. In this paper, we study a reconfiguration variant for maximum-weight b-matchings: For two given maximum-weight b-matchings in a graph, we are asked to determine whether there exists a sequence of maximum-weight b-matchings in the graph between them, with subsequent b-matchings obtained by removing one edge and adding another. We show that this reconfiguration problem is solvable in polynomial time for instances with no integrality gap. Such instances include bipartite graphs with any capacity function on vertices, and 2-matchings in general graphs. Thus, our result implies that the reconfiguration problem for maximum-weight matchings can be solved in polynomial time for bipartite graphs. We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as @math -free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time. Given the prevalence of social media websites, one challenge facing computer vision researchers is to devise methods to search for persons of interest among the billions of shared photos on these websites. Despite significant progress in face recognition, searching a large collection of unconstrained face images remains a difficult problem. To address this challenge, we propose a face search system which combines a fast search procedure, coupled with a state-of-the-art commercial off the shelf (COTS) matcher, in a cascaded framework. Given a probe face, we first filter the large gallery of photos to find the top- @math most similar faces using features learned by a convolutional neural network. The @math retrieved candidates are re-ranked by combining similarities based on deep features and those output by the COTS matcher. We evaluate the proposed face search system on a gallery containing @math million web-downloaded face images. Experimental results demonstrate that while the deep features perform worse than the COTS matcher on a mugshot dataset (93.7 percent versus 98.6 percent TAR@FAR of 0.01 percent), fusing the deep features with the COTS matcher improves the overall performance ( @math percent TAR@FAR of 0.01 percent). This shows that the learned deep features provide complementary information over representations used in state-of-the-art face matchers. On the unconstrained face image benchmarks, the performance of the learned deep features is competitive with reported accuracies. LFW database: @math percent accuracy under the standard protocol and @math percent TAR@FAR of @math percent under the BLUFR protocol; IJB-A benchmark: @math percent TAR@FAR of @math percent (verification), rank 1 retrieval of @math percent (closed-set search), @math percent FNIR@FAR of @math percent (open-set search). The proposed face search system offers an excellent trade-off between accuracy and scalability on galleries with millions of images. Additionally, in a face search experiment involving photos of the Tsarnaev brothers, convicted of the Boston Marathon bombing, the proposed cascade face search system could find the younger brother's (Dzhokhar Tsarnaev) photo at rank @math in @math second on a @math M gallery and at rank @math in @math seconds on an @math M gallery. We present techniques to parallelize membership tests for Deterministic Finite Automata (DFAs). Our method searches arbitrary regular expressions by matching multiple bytes in parallel using speculation. We partition the input string into chunks, match chunks in parallel, and combine the matching results. Our parallel matching algorithm exploits structural DFA properties to minimize the speculative overhead. Unlike previous approaches, our speculation is failure-free, i.e., (1) sequential semantics are maintained, and (2) speed-downs are avoided altogether. On architectures with a SIMD gather-operation for indexed memory loads, our matching operation is fully vectorized. The proposed load-balancing scheme uses an off-line profiling step to determine the matching capacity of each participating processor. Based on matching capacities, DFA matches are load-balanced on inhomogeneous parallel architectures such as cloud computing environments. We evaluated our speculative DFA membership test for a representative set of benchmarks from the Perl-compatible Regular Expression (PCRE) library and the PROSITE protein database. Evaluation was conducted on a 4 CPU (40 cores) shared-memory node of the Intel Academic Program Manycore Testing Lab (Intel MTL), on the Intel AVX2 SDE simulator for 8-way fully vectorized SIMD execution, and on a 20-node (288 cores) cluster on the Amazon EC2 computing cloud. Obtained speedups are on the order of @math O 1 + | P | - 1 | Q | · ? , where @math | P | denotes the number of processors or SIMD units, @math | Q | denotes the number of DFA states, and @math 0 < ? ≤ 1 represents a statically computed DFA property. For all observed cases, we found that @math 0.02 < ? < 0.47 . Actual speedups range from 2.3 @math × to 38.8 @math × for up to 512 DFA states for PCRE, and between 1.3 @math × and 19.9 @math × for up to 1,288 DFA states for PROSITE on a 40-core MTL node. Speedups on the EC2 computing cloud range from 5.0 @math × to 65.8 @math × for PCRE, and from 5.0 @math × to 138.5 @math × for PROSITE. Speedups of our C-based DFA matcher over the Perl-based ScanProsite scan tool range from 559.3 @math × to 15079.7 @math × on a 40-core MTL node. We show the scalability of our approach for input-sizes of up to 10 GB.