Approximating submodular kk-partition via principal partition sequence

In submodular $k$-partition, the input is a non-negative submodular function $f$ defined over a finite ground set $V$ (given by an evaluation oracle) along with a positive integer $k$ and the goal is to find a partition of the ground set $V$ into $k$ non-empty parts $V_1, V_2, ..., V_k$ in order to minimize $\sum_{i=1}^k f(V_i)$. Narayanan, Roy, and Patkar (Journal of Algorithms, 1996) designed an algorithm for submodular $k$-partition based on the principal partition sequence and showed that the approximation factor of their algorithm is $2$ for the special case of graph cut functions (subsequently rediscovered by Ravi and Sinha (Journal of Operational Research, 2008)). In this work, we study the approximation factor of their algorithm for three subfamilies of submodular functions -- monotone, symmetric, and posimodular, and show the following results: 1. The approximation factor of their algorithm for monotone submodular $k$-partition is $4/3$. This result improves on the $2$-factor achievable via other algorithms. Moreover, our upper bound of $4/3$ matches the recently shown lower bound under polynomial number of function evaluation queries (Santiago, IWOCA 2021). Our upper bound of $4/3$ is also the first improvement beyond $2$ for a certain graph partitioning problem that is a special case of monotone submodular $k$-partition. 2. The approximation factor of their algorithm for symmetric submodular $k$-partition is $2$. This result generalizes their approximation factor analysis beyond graph cut functions. 3. The approximation factor of their algorithm for posimodular submodular $k$-partition is $2$. We also construct an example to show that the approximation factor of their algorithm for arbitrary submodular functions is $\Omega(n/k)$.Comment: Accepted to APPROX'2

Counting and enumerating optimum cut sets for hypergraph kk-partitioning problems for fixed kk

We consider the problem of enumerating optimal solutions for two hypergraph $k$-partitioning problems -- namely, Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. The input in hypergraph $k$-partitioning problems is a hypergraph $G=(V, E)$ with positive hyperedge costs along with a fixed positive integer $k$. The goal is to find a partition of $V$ into $k$ non-empty parts $(V_1, V_2, \ldots, V_k)$ -- known as a $k$-partition -- so as to minimize an objective of interest. 1. If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-$k$-Partition. A subset of hyperedges is a minmax-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Minmax-Hypergraph-$k$-Partition. 2. If the objective of interest is the total cost of hyperedges crossing the $k$-partition, then the problem is known as Hypergraph-$k$-Cut. A subset of hyperedges is a min-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Hypergraph-$k$-Cut. We give the first polynomial bound on the number of minmax-$k$-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed $k$. Our technique is strong enough to also enable an $n^{O(k)}p$-time deterministic algorithm to enumerate all min-$k$-cut-sets in hypergraphs, thus improving on the previously known $n^{O(k^2)}p$-time deterministic algorithm, where $n$ is the number of vertices and $p$ is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).Comment: Accepted to ICALP'22. arXiv admin note: text overlap with arXiv:2110.1481

SketchFFusion: Sketch-guided image editing with diffusion model

Sketch-guided image editing aims to achieve local fine-tuning of the image based on the sketch information provided by the user, while maintaining the original status of the unedited areas. Due to the high cost of acquiring human sketches, previous works mostly relied on edge maps as a substitute for sketches, but sketches possess more rich structural information. In this paper, we propose a sketch generation scheme that can preserve the main contours of an image and closely adhere to the actual sketch style drawn by the user. Simultaneously, current image editing methods often face challenges such as image distortion, training cost, and loss of fine details in the sketch. To address these limitations, We propose a conditional diffusion model (SketchFFusion) based on the sketch structure vector. We evaluate the generative performance of our model and demonstrate that it outperforms existing methods

PaReNTT: Low-Latency Parallel Residue Number System and NTT-Based Long Polynomial Modular Multiplication for Homomorphic Encryption

High-speed long polynomial multiplication is important for applications in homomorphic encryption (HE) and lattice-based cryptosystems. This paper addresses low-latency hardware architectures for long polynomial modular multiplication using the number-theoretic transform (NTT) and inverse NTT (iNTT). Chinese remainder theorem (CRT) is used to decompose the modulus into multiple smaller moduli. Our proposed architecture, namely PaReNTT, makes four novel contributions. First, parallel NTT and iNTT architectures are proposed to reduce the number of clock cycles to process the polynomials. This can enable real-time processing for HE applications, as the number of clock cycles to process the polynomial is inversely proportional to the level of parallelism. Second, the proposed architecture eliminates the need for permuting the NTT outputs before their product is input to the iNTT. This reduces latency by n/4 clock cycles, where n is the length of the polynomial, and reduces buffer requirement by one delay-switch-delay circuit of size n. Third, an approach to select special moduli is presented where the moduli can be expressed in terms of a few signed power-of-two terms. Fourth, novel architectures for pre-processing for computing residual polynomials using the CRT and post-processing for combining the residual polynomials are proposed. These architectures significantly reduce the area consumption of the pre-processing and post-processing steps. The proposed long modular polynomial multiplications are ideal for applications that require low latency and high sample rate as these feed-forward architectures can be pipelined at arbitrary levels

A multifunctional ribonuclease A-conjugated carbon dot cluster nanosystem for synchronous cancer imaging and therapy

Carbon dots exhibit great potential in applications such as molecular imaging and in vivo molecular tracking. However, how to enhance fluorescence intensity of carbon dots has become a great challenge. Herein, we report for the first time a new strategy to synthesize fluorescent carbon dots (C-dots) with high quantum yields by using ribonuclease A (RNase A) as a biomolecular templating agent under microwave irradiation. The synthesized RNase A-conjugated carbon dots (RNase A@C-dots) exhibited quantum yields of 24.20%. The fluorescent color of the RNase A@C-dots can easily be adjusted by varying the microwave reaction time and microwave power. Moreover, the emission wavelength and intensity of RNase A@C-dots displayed a marked excitation wavelength-dependent character. As the excitation wavelength alters from 300 to 500 nm, the photoluminescence (PL) peak exhibits gradually redshifts from 450 to 550 nm, and the intensity reaches its maximum at an excitation wavelength of 380 nm. Its Stokes shift is about 80 nm. Notably, the PL intensity is gradually decreasing as the pH increases, almost linearly dependent, and it reaches the maximum at a pH = 2 condition; the emission peaks also show clearly a redshift, which may be caused by the high activity and perfective dispersion of RNase A in a lower pH solution. In high pH solution, RNase A tends to form RNase A warped carbon dot nanoclusters. Cell imaging confirmed that the RNase A@C-dots could enter into the cytoplasm through cell endocytosis. 3D confocal imaging and transmission electron microscopy observation confirmed partial RNase A@C-dots located inside the nucleus. MTT and real-time cell electronic sensing (RT-CES) analysis showed that the RNase A@C-dots could effectively inhibit the growth of MGC-803 cells. Intra-tumor injection test of RNase A@C-dots showed that RNase A@C-dots could be used for imaging in vivo gastric cancer cells. In conclusion, the as-prepared RNase A@C-dots are suitable for simultaneous therapy and in vivo fluorescence imaging of nude mice loaded with gastric cancer or other tumors

Soliton generation in CaF2_2 crystalline whispering gallery mode resonators with negative thermal-optical effects

Calcium fluoride (CaF$_2$) crystalline whispering gallery mode resonators (WGMRs) exhibit ultrahigh intrinsic quality factors and a low power anomalous dispersion in the communication and mid-infrared bands, making them attractive platforms for microresonator-based comb generation. However, their unique negative thermo-optic effects pose challenges when achieving thermal equilibrium. To our knowledge, our experiments serve as the first demonstration of soliton microcombs in Q > 109 CaF$_2$ WGMRs. We observed soliton mode-locking and bidirectional switching of soliton numbers caused by the negative thermo-optic effects. Additionally, various soliton formation dynamics are shown, including breathing and vibrational solitons, which can be attributed to thermo-photomechanical oscillations. Thus, our results enrich the soliton generation platform and provide a reference for generating solitons from WGMRs that comprise other materials with negative thermo-optic effects. In the future, the ultrahigh quality factor of CaF$_2$ crystal cavities may enable the generation of sub-milliwatt-level broad-spectrum soliton combs.Comment: 4 pages,5 pictures,description of soliton generation in a calcium fluoride whisper gallery mode microresonators with negative thermo-optical effect,ready for publication in optics lette