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We give the first almost-linear total time algorithm for deciding if a flow of cost at most $F$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and $s$-$t$ distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $m^{1+o(1)}$ dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $m^{o(1)}$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-R\"{a}cke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

Man\v{c}inska and Roberson~[FOCS'20] showed that two graphs are quantum isomorphic if and only if they are homomorphism indistinguishable over the class of planar graphs. Atserias et al.~[JCTB'19] proved that quantum isomorphism is undecidable in general. The NPA hierarchy gives a sequence of semidefinite programming relaxations of quantum isomorphism. Recently, Roberson and Seppelt~[ICALP'23] obtained a homomorphism indistinguishability characterization of the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism. We prove a quantum analogue of this result by showing that each level of the NPA hierarchy of SDP relaxations for quantum isomorphism of graphs is equivalent to homomorphism indistinguishability over an appropriate class of planar graphs. By combining the convergence of the NPA hierarchy with the fact that the union of these graph classes is the set of all planar graphs, we are able to give a new proof of the result of Man\v{c}inska and Roberson~[FOCS'20] that avoids the use of the theory of quantum groups. This homomorphism indistinguishability characterization also allows us to give a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.

We first establish the unique ergodicity of the Markov chain generated by the stochastic theta method (STM) with $\theta \in [1/2, 1]$ for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and $\theta$, and the irreducibility and the strong Feller property for the STM. We then generalize the arguments to the temporal drift-implicit Euler (DIE) method and its Galerkin-based full discretizations for a class of monotone SPDEs driven by infinite-dimensional nondegenerate multiplicative trace-class noise. Applying these results to the stochastic Allen--Cahn equation indicates that its DIE scheme is uniquely ergodic for any interface thickness, which gives an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93). Numerical experiments verify our theoretical results.

We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in $\mathcal{O}(k \cdot n^{1-1/d} \cdot m \cdot \mathrm{polylog}(n))$ time complexity for the graph on $n$ vertices and $m$ edges, where $k$ is the diameter and $d$ is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an $\mathcal{O}(n^{7/4} \cdot \mathrm{polylog}(n))$ parameterized diameter algorithm for unit square intersection graph of size $n$, as well as a more general algorithm for convex polygon intersection graphs.

In 2022, Chen et al. proposed an algorithm in \cite{main} that solves the min cost flow problem in $m^{1 + o(1)} \log U \log C$ time, where $m$ is the number of edges in the graph, $U$ is an upper bound on capacities and $C$ is an upper bound on costs. However, as far as the authors of \cite{main} know, no one has implemented their algorithm to date. In this paper, we discuss implementations of several key portions of the algorithm given in \cite{main}, including the justifications for specific implementation choices. For the portions of the algorithm that we do not implement, we provide stubs. We then go through the entire algorithm and calculate the $m^{o(1)}$ term more precisely. Finally, we conclude with potential directions for future work in this area.

We consider a weakly supervised learning scenario where the supervision signal is generated by a transition function $\sigma$ of labels associated with multiple input instances. We formulate this problem as \emph{multi-instance Partial Label Learning (multi-instance PLL)}, which is an extension to the standard PLL problem. Our problem is met in different fields, including latent structural learning and neuro-symbolic integration. Despite the existence of many learning techniques, limited theoretical analysis has been dedicated to this problem. In this paper, we provide the first theoretical study of multi-instance PLL with possibly an unknown transition $\sigma$. Our main contributions are as follows. Firstly, we propose a necessary and sufficient condition for the learnability of the problem. This condition non-trivially generalizes and relaxes the existing small ambiguity degree in the PLL literature, since we allow the transition to be deterministic. Secondly, we derive Rademacher-style error bounds based on a top-$k$ surrogate loss that is widely used in the neuro-symbolic literature. Furthermore, we conclude with empirical experiments for learning under unknown transitions. The empirical results align with our theoretical findings; however, they also expose the issue of scalability in the weak supervision literature.

The all-pairs shortest distances (APSD) with differential privacy (DP) problem takes as input an undirected, weighted graph $G = (V,E, \mathbf{w})$ and outputs a private estimate of the shortest distances in $G$ between all pairs of vertices. In this paper, we present a simple $\widetilde{O}(n^{1/3}/\varepsilon)$-accurate algorithm to solve APSD with $\varepsilon$-DP, which reduces to $\widetilde{O}(n^{1/4}/\varepsilon)$ in the $(\varepsilon, \delta)$-DP setting, where $n = |V|$. Our algorithm greatly improves upon the error of prior algorithms, namely $\widetilde{O}(n^{2/3}/\varepsilon)$ and $\widetilde{O}(\sqrt{n}/\varepsilon)$ in the two respective settings, and is the first to be optimal up to a polylogarithmic factor, based on a lower bound of $\widetilde{\Omega}(n^{1/4})$. In the case where a multiplicative approximation is allowed, we give two different constructions of algorithms with reduced additive error. Our first construction allows a multiplicative approximation of $O(k\log{\log{n}})$ and has additive error $\widetilde{O}(k\cdot n^{1/k}/\varepsilon)$ in the $\varepsilon$-DP case and $\widetilde{O}(\sqrt{k}\cdot n^{1/(2k)}/\varepsilon)$ in the $(\varepsilon, \delta)$-DP case. Our second construction allows multiplicative approximation $2k-1$ and has the same asymptotic additive error as the first construction. Both constructions significantly improve upon the currently best-known additive error of, $\widetilde{O}(k\cdot n^{1/2 + 1/(4k+2)}/\varepsilon)$ and $\widetilde{O}(k\cdot n^{1/3 + 2/(9k+3)}/\varepsilon)$, respectively. Our algorithms are straightforward and work by decomposing a graph into a set of spanning trees, and applying a key observation that we can privately release APSD in trees with $O(\text{polylog}(n))$ error.

We consider the Max Unique Coverage problem, including applications to the data stream model. The input is a universe of $n$ elements, a collection of $m$ subsets of this universe, and a cardinality constraint, $k$. The goal is to select a subcollection of at most $k$ sets that maximizes unique coverage, i.e, the number of elements contained in exactly one of the selected sets. The Max Unique Coverage problem has applications in wireless networks, radio broadcast, and envy-free pricing. Our first main result is a fixed-parameter tractable approximation scheme (FPT-AS) for Max Unique Coverage, parameterized by $k$ and the maximum element frequency, $r$, which can be implemented on a data stream. Our FPT-AS finds a $(1-\epsilon)$-approximation while maintaining a kernel of size $\tilde{O}(k r/\epsilon)$, which can be combined with subsampling to use $\tilde{O}(k^2 r / \epsilon^3)$ space overall. This significantly improves on the previous-best FPT-AS with the same approximation, but a kernel of size $\tilde{O}(k^2 r / \epsilon^2)$. In order to achieve our result, we show upper bounds on the ratio of a collection's coverage to the unique coverage of a maximizing subcollection; this is by constructing explicit algorithms that find a subcollection with unique coverage at least a logarithmic ratio of the collection's coverage. We complement our algorithms with our second main result, showing that $\Omega(m / k^2)$ space is necessary to achieve a $(1.5 + o(1))/(\ln k - 1)$-approximation in the data stream. This dramatically improves the previous-best lower bound showing that $\Omega(m / k^2)$ is necessary to achieve better than a $e^{-1+1/k}$-approximation.

Since the work of Polyanskiy, Poor and Verd\'u on the finite blocklength performance of capacity-achieving codes for discrete memoryless channels, many papers have attempted to find further results for more practically relevant channels. However, it seems that the complexity of computing capacity-achieving codes has not been investigated until now. We study this question for the simplest non-trivial Gaussian channels, i.e., the additive colored Gaussian noise channel. To assess the computational complexity, we consider the classes $\mathrm{FP}_1$ and $\#\mathrm{P}_1$. $\mathrm{FP}_1$ includes functions computable by a deterministic Turing machine in polynomial time, whereas $\#\mathrm{P}_1$ encompasses functions that count the number of solutions verifiable in polynomial time. It is widely assumed that $\mathrm{FP}_1\neq\#\mathrm{P}_1$. It is of interest to determine the conditions under which, for a given $M \in \mathbb{N}$, where $M$ describes the precision of the deviation of $C(P,N)$, for a certain blocklength $n_M$ and a decoding error $\epsilon > 0$ with $\epsilon\in\mathbb{Q}$, the following holds: $R_{n_M}(\epsilon)>C(P,N)-\frac{1}{2^M}$. It is shown that there is a polynomial-time computable $N_*$ such that for sufficiently large $P_*\in\mathbb{Q}$, the sequences $\{R_{n_M}(\epsilon)\}_{{n_M}\in\mathbb{N}}$, where each $R_{n_M}(\epsilon)$ satisfies the previous condition, cannot be computed in polynomial time if $\mathrm{FP}_1\neq\#\mathrm{P}_1$. Hence, the complexity of computing the sequence $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$ grows faster than any polynomial as $M$ increases. Consequently, it is shown that either the sequence of achievable rates $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$ as a function of the blocklength, or the sequence of blocklengths $\{n_M\}_{M\in\mathbb{N}}$ corresponding to the achievable rates, is not a polynomial-time computable sequence.

In multi-turn dialog, utterances do not always take the full form of sentences \cite{Carbonell1983DiscoursePA}, which naturally makes understanding the dialog context more difficult. However, it is essential to fully grasp the dialog context to generate a reasonable response. Hence, in this paper, we propose to improve the response generation performance by examining the model's ability to answer a reading comprehension question, where the question is focused on the omitted information in the dialog. Enlightened by the multi-task learning scheme, we propose a joint framework that unifies these two tasks, sharing the same encoder to extract the common and task-invariant features with different decoders to learn task-specific features. To better fusing information from the question and the dialog history in the encoding part, we propose to augment the Transformer architecture with a memory updater, which is designed to selectively store and update the history dialog information so as to support downstream tasks. For the experiment, we employ human annotators to write and examine a large-scale dialog reading comprehension dataset. Extensive experiments are conducted on this dataset, and the results show that the proposed model brings substantial improvements over several strong baselines on both tasks. In this way, we demonstrate that reasoning can indeed help better response generation and vice versa. We release our large-scale dataset for further research.