We present representative sets-style statements for linear delta-matroids, which are set systems that generalize matroids, with important connections to matching theory and graph embeddings. Furthermore, our proof uses a new approach of sieving polynomial families, which generalizes the linear algebra approach of the representative sets lemma to a setting of bounded-degree polynomials. The representative sets statements for linear delta-matroids then follow by analyzing the Pfaffian of the skew-symmetric matrix representing the delta-matroid. Applying the same framework to the determinant instead of the Pfaffian recovers the representative sets lemma for linear matroids. Altogether, this significantly extends the toolbox available for kernelization. As an application, we show an exact sparsification result for Mader networks: Let $G=(V,E)$ be a graph and $\mathcal{T}$ a partition of a set of terminals $T \subseteq V(G)$, $|T|=k$. A $\mathcal{T}$-path in $G$ is a path with endpoints in distinct parts of $\mathcal{T}$ and internal vertices disjoint from $T$. In polynomial time, we can derive a graph $G'=(V',E')$ with $T \subseteq V(G')$, such that for every subset $S \subseteq T$ there is a packing of $\mathcal{T}$-paths with endpoints $S$ in $G$ if and only if there is one in $G'$, and $|V(G')|=O(k^3)$. This generalizes the (undirected version of the) cut-covering lemma, which corresponds to the case that $\mathcal{T}$ contains only two blocks. To prove the Mader network sparsification result, we furthermore define the class of Mader delta-matroids, and show that they have linear representations. This should be of independent interest.
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An NP-complete graph decision problem, the "Multi-stage graph Simple Path" (abbr. MSP) problem, is introduced. The main contribution of this paper is a poly-time algorithm named the ZH algorithm for the problem together with the proof of its correctness, which implies NP=P. (1) A crucial structural property is discovered, whereby all MSP instances are arranged into the sequence $G_{0}$,$G_{1}$,$G_{2}$,... ($G_{k}$ essentially stands for a group of graphs for all $k\geq 0$). For each $G_{j}(j>0)$ in the sequence, there is a graph $G_{i}(0\leq i<j)$ "mathematically homomorphic" to $G_{j}$ which keeps completely accordant with $G_{j}$ on the existence of global solutions. This naturally provides a chance of applying mathematical induction for the proof of an algorithm. In previous attempts, algorithms used for making global decisions were mostly guided by heuristics and intuition. Rather, the ZH algorithm is dedicatedly designed to comply with the proposed proving framework of mathematical induction. (2) Although the ZH algorithm deals with paths, it always regards paths as a collection of edge sets. This is the key to the avoidance of exponential complexity. (3) Any poly-time algorithm that seeks global information can barely avoid the error caused by localized computation. In the ZH algorithm, the proposed reachable-path edge-set $R(e)$ and the computed information for it carry all necessary contextual information, which can be utilized to summarize the "history" and to detect the "future" for searching global solutions. (4) The relation between local strategies and global strategies is discovered and established, wherein preceding decisions can pose constraints to subsequent decisions (and vice versa). This interplay resembles the paradigm of dynamic programming, while much more convoluted. Nevertheless, the computation is always strait forward and decreases monotonically.
This paper investigates the problem of efficient constrained global optimization of hybrid models that are a composition of a known white-box function and an expensive multi-output black-box function subject to noisy observations, which often arises in real-world science and engineering applications. We propose a novel method, Constrained Upper Quantile Bound (CUQB), to solve such problems that directly exploits the composite structure of the objective and constraint functions that we show leads substantially improved sampling efficiency. CUQB is a conceptually simple, deterministic approach that avoid constraint approximations used by previous methods. Although the CUQB acquisition function is not available in closed form, we propose a novel differentiable sample average approximation that enables it to be efficiently maximized. We further derive bounds on the cumulative regret and constraint violation under a non-parametric Bayesian representation of the black-box function. Since these bounds depend sublinearly on the number of iterations under some regularity assumptions, we establis bounds on the convergence rate to the optimal solution of the original constrained problem. In contrast to most existing methods, CUQB further incorporates a simple infeasibility detection scheme, which we prove triggers in a finite number of iterations when the original problem is infeasible (with high probability given the Bayesian model). Numerical experiments on several test problems, including environmental model calibration and real-time optimization of a reactor system, show that CUQB significantly outperforms traditional Bayesian optimization in both constrained and unconstrained cases. Furthermore, compared to other state-of-the-art methods that exploit composite structure, CUQB achieves competitive empirical performance while also providing substantially improved theoretical guarantees.
We investigate the fixed-budget best-arm identification (BAI) problem for linear bandits in a potentially non-stationary environment. Given a finite arm set $\mathcal{X}\subset\mathbb{R}^d$, a fixed budget $T$, and an unpredictable sequence of parameters $\left\lbrace\theta_t\right\rbrace_{t=1}^{T}$, an algorithm will aim to correctly identify the best arm $x^* := \arg\max_{x\in\mathcal{X}}x^\top\sum_{t=1}^{T}\theta_t$ with probability as high as possible. Prior work has addressed the stationary setting where $\theta_t = \theta_1$ for all $t$ and demonstrated that the error probability decreases as $\exp(-T /\rho^*)$ for a problem-dependent constant $\rho^*$. But in many real-world $A/B/n$ multivariate testing scenarios that motivate our work, the environment is non-stationary and an algorithm expecting a stationary setting can easily fail. For robust identification, it is well-known that if arms are chosen randomly and non-adaptively from a G-optimal design over $\mathcal{X}$ at each time then the error probability decreases as $\exp(-T\Delta^2_{(1)}/d)$, where $\Delta_{(1)} = \min_{x \neq x^*} (x^* - x)^\top \frac{1}{T}\sum_{t=1}^T \theta_t$. As there exist environments where $\Delta_{(1)}^2/ d \ll 1/ \rho^*$, we are motivated to propose a novel algorithm $\mathsf{P1}$-$\mathsf{RAGE}$ that aims to obtain the best of both worlds: robustness to non-stationarity and fast rates of identification in benign settings. We characterize the error probability of $\mathsf{P1}$-$\mathsf{RAGE}$ and demonstrate empirically that the algorithm indeed never performs worse than G-optimal design but compares favorably to the best algorithms in the stationary setting.
Coded distributed computing, proposed by Li et al., offers significant potential for reducing the communication load in MapReduce computing systems. In the setting of the \emph{cascaded} coded distributed computing that consisting of $K$ nodes, $N$ input files, and $Q$ output functions, the objective is to compute each output function through $s\geq 1$ nodes with a computation load $r\geq 1$, enabling the application of coding techniques during the Shuffle phase to achieve minimum communication load. However, for most existing coded distributed computing schemes, a major limitation lies in their demand for splitting the original data into an exponentially growing number of input files in terms of $N/\binom{K}{r} \in\mathbb{N}$ and requiring an exponentially large number of output functions $Q/\binom{K}{s} \in\mathbb{N}$, which imposes stringent requirements for implementation and results in significant coding complexity when $K$ is large. In this paper, we focus on the cascaded case of $K/s\in\mathbb{N} $, deliberately designing the strategy of input files store and output functions assignment based on a grouping method, such that a low-complexity two-round Shuffle phase is available. The main advantages of our proposed scheme contains: 1) the communication load is quilt close to or surprisingly better than the optimal state-of-the-art scheme proposed by Li et al.; 2) our scheme requires significantly less number of input files and output functions; 3) all the operations are implemented over the minimum binary field $\mathbb{F}_2$.
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a $2$-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a simple rounding algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair $(G, S)$, consisting of a graph with an odd cycle transversal. If $S$ is a stable set, we prove a tight approximation ratio of $1 + 1/\rho$, where $2\rho -1$ denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph $\tilde{G} := G /S$ and satisfies $\rho \in [2,\infty]$. If $S$ is an arbitrary set, we prove a tight approximation ratio of $\left(1+1/\rho \right) (1 - \alpha) + 2 \alpha$, where $\alpha \in [0,1]$ is a natural parameter measuring the quality of the set $S$. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph $\tilde{G}$. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals and show optimality of the analysis.
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity of a composition of functions $f\diamond g$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^{1}$. The intuition that underlies the KRW conjecture is that the composition $f\diamond g$ should behave like a "direct-sum problem", in a certain sense, and therefore the depth complexity of $f\diamond g$ should be the sum of the individual depth complexities. Nevertheless, there are two obstacles toward turning this intuition into a proof: first, we do not know how to prove that $f\diamond g$ must behave like a direct-sum problem; second, we do not know how to prove that the complexity of the latter direct-sum problem is indeed the sum of the individual complexities. In this work, we focus on the second obstacle. To this end, we study a notion called "strong composition", which is the same as $f\diamond g$ except that it is forced to behave like a direct-sum problem. We prove a variant of the KRW conjecture for strong composition, thus overcoming the above second obstacle. This result demonstrates that the first obstacle above is the crucial barrier toward resolving the KRW conjecture. Along the way, we develop some general techniques that might be of independent interest.
The Weighted Path Order of Yamada is a powerful technique for proving termination. It is also supported by CeTA, a certifier for checking untrusted termination proofs. To be more precise, CeTA contains a verified function that computes for two terms whether one of them is larger than the other for a given WPO, i.e., where all parameters of the WPO have been fixed. The problem of this verified function is its exponential runtime in the worst case. Therefore, in this work we develop a polynomial time implementation of WPO that is based on memoization. It also improves upon an earlier verified implementation of the Recursive Path Order: the RPO-implementation uses full terms as keys for the memory, a design which simplified the soundness proofs, but has some runtime overhead. In this work, keys are just numbers, so that the lookup in the memory is faster. Although trivial on paper, this change introduces some challenges for the verification task.
We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of $n$ disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter $r$. The case of intersection graphs is a special case with $r=0$. We give an algorithm that, given a target length $k$, computes the smallest value of $r$ for which there is a path of length at most $k$ between some given pair of disks in the proximity graph. Our algorithm runs in $O^*(n^{5/4})$ randomized expected time, which improves to $O^*(n^{6/5})$ for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and $k$ is replaced by a target weight $w$; that is, we seek a path whose length is at most $w$. In other variants, we want to optimize a parameter different from $r$, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given $r$ has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra's algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [4].
We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. This is the first work in which we study the dual codes in the framework of the two generic constructions; in particular, we propose a Gram-Schmidt (complexity of $\mathcal{O}(n^3)$) process to compute them explicitly. The originality of this contribution is in the study of the existence or not of defining sets $D'$, which can be used as ingredients to construct the dual code $\mathcal{C}'$ for a given code $\mathcal{C}$ in the context of the second generic construction. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of $\mathcal{C}$ to belong in the dual. This achievement was done in general and when the involved functions are weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals.
Gradient Descent (GD) is a powerful workhorse of modern machine learning thanks to its scalability and efficiency in high-dimensional spaces. Its ability to find local minimisers is only guaranteed for losses with Lipschitz gradients, where it can be seen as a `bona-fide' discretisation of an underlying gradient flow. Yet, many ML setups involving overparametrised models do not fall into this problem class, which has motivated research beyond the so-called ``Edge of Stability'' (EoS), where the step-size crosses the admissibility threshold inversely proportional to the Lipschitz constant above. Perhaps surprisingly, GD has been empirically observed to still converge regardless of local instability and oscillatory behavior. The incipient theoretical analysis of this phenomena has mainly focused in the overparametrised regime, where the effect of choosing a large learning rate may be associated to a `Sharpness-Minimisation' implicit regularisation within the manifold of minimisers, under appropriate asymptotic limits. In contrast, in this work we directly examine the conditions for such unstable convergence, focusing on simple, yet representative, learning problems, via analysis of two-step gradient updates. Specifically, we characterize a local condition involving third-order derivatives that guarantees existence and convergence to fixed points of the two-step updates, and leverage such property in a teacher-student setting, under population loss. Finally, starting from Matrix Factorization, we provide observations of period-2 orbit of GD in high-dimensional settings with intuition of its dynamics, along with exploration into more general settings.
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