arXiv subjects

Shengminjie Chen

Publications and source records attributed to Shengminjie Chen.

6 recordsLinked to original sources

arXiv2026

Breaking the 1/3 Barrier for $\boldsymbol{k}$-Submodular Maximization under Matroid and Knapsack Constraints: A Proportional Top-2 Randomized Framework

$k$-submodularity generalizes submodularity by allowing each selected element to be assigned one of $k$ labels, rather than being merely selected or not selected. We study the problem of maximizing a nonnegative non-monotone $k$-submodular function, where $k\ge 2$, under classical support constraints, including a single matroid constraint and a single knapsack constraint. Previously, the best known approximation guarantees for non-monotone constrained $k$-submodular maximization had long remained at $1/3$ or $1/3-\varepsilon$, even in basic settings such as cardinality, matroid, and knapsack constraints. We show that this $1/3$ barrier is not inherent: for both the matroid and knapsack settings considered here, we give randomized polynomial-time algorithms achieving an approximation ratio of $\sqrt{2}-1\approx 0.4142$. The algorithms use a simple randomized greedy rule: once an element is selected, its label is chosen only from the two labels with the largest marginal gains, with probabilities proportional to the positive parts of these two gains. The value-oracle query complexity is $O(n^2k)$ in the matroid setting and $O(n^3k^2)$ in the knapsack setting. These results give the first approximation guarantees exceeding $1/3$ for non-monotone $k$-submodular maximization under matroid and knapsack constraints.

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arXiv2026

Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

Submodular maximization constitutes a prominent research topic in combinatorial optimization and theoretical computer science, with extensive applications across diverse domains. While substantial advancements have been achieved in approximation algorithms for submodular maximization, the majority of algorithms yielding high approximation guarantees are randomized. In this work, we investigate deterministic approximation algorithms for maximizing non-monotone submodular functions subject to matroid and knapsack constraints. For the two distinct constraint settings, we propose novel deterministic algorithms grounded in an extended multilinear extension framework. Under matroid constraints, our algorithm achieves an approximation ratio of $(0.385 - ε)$, whereas for knapsack constraints, the proposed algorithm attains an approximation ratio of $(0.367 -ε)$. Both algorithms run in $\mathrm{poly}(n)$ query complexity, where $n$ is the size of the ground set, and improve upon the state-of-the-art deterministic approximation ratios of $(0.367 - ε)$ for matroid constraints and $0.25$ for knapsack constraints.

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arXiv2025

A Lyapunov Framework for Quantum Algorithm Design in Combinatorial Optimization with Approximation Ratio Guarantees

In this work, we develop a framework aiming at designing quantum algorithms for combinatorial optimization problems while providing theoretical guarantees on their approximation ratios. The principal innovative aspect of our work is the construction of a time-dependent Lyapunov function that naturally induces a controlled Schrödinger evolution with a time dependent Hamiltonian for maximizing approximation ratios of algorithms. Because the approximation ratio depends on the optimal solution, which is typically elusive and difficult to ascertain a priori, the second novel component is to construct the upper bound of the optimal solution through the current quantum state. By enforcing the non-decreasing property of this Lyapunov function, we not only derive a class of quantum dynamics that can be simulated by quantum devices but also obtain rigorous bounds on the achievable approximation ratio. As a concrete demonstration, we apply our framework to Max-Cut problem, implementing it as an adaptive variational quantum algorithm based on a Hamiltonian ansatz. This algorithm avoids ansatz and graph structural assumptions and bypasses parameter training through a tunable parameter function integrated with measurement feedback.

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arXiv2025

A Unified Complexity-Algorithm Account of Constant-Round QAOA Expectation Computation

The Quantum Approximate Optimization Algorithm (QAOA) is widely studied for combinatorial optimization and has achieved significant advances both in theoretical guarantees and practical performance, yet for general combinatorial optimization problems the expected performance and classical simulability of fixed-round QAOA remain unclear. Focusing on Max-Cut, we first show that for general graphs and any fixed round $p\ge2$, exactly evaluating the expectation of fixed-round QAOA at prescribed angles is $\mathrm{NP}$-hard, and that approximating this expectation within additive error $2^{-O(n)}$ in the number $n$ of vertices is already $\mathrm{NP}$-hard. To evaluate the expected performance of QAOA, we propose a dynamic programming algorithm leveraging tree decomposition. As a byproduct, when the $p$-local treewidth grows at most logarithmically with the number of vertices, this yields a polynomial-time \emph{exact} evaluation algorithm in the graph size $n$. Beyond Max-Cut, we extend the framework to general Binary Unconstrained Combinatorial Optimization (BUCO). Finally, we provide reproducible evaluations for rounds up to $p=3$ on representative structured families, including the generalized Petersen graph $GP(15,2)$, double-layer triangular 2-lifts, and the truncated icosahedron graph $C_{60}$, and report cut ratios while benchmarking against locality-matched classical baselines.

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arXiv2025

Stochastic Quantum Hamiltonian Descent

Stochastic Gradient Descent (SGD) and its variants underpin modern machine learning by enabling efficient optimization of large-scale models. However, their local search nature limits exploration in complex landscapes. In this paper, we introduce Stochastic Quantum Hamiltonian Descent (SQHD), a quantum optimization algorithm that integrates the computational efficiency of stochastic gradient methods with the global exploration power of quantum dynamics. We propose a Lindbladian dynamics as the quantum analogue of continuous-time SGD. We further propose a discrete-time gate-based algorithm that approximates these dynamics while avoiding direct Lindbladian simulation, enabling practical implementation on near-term quantum devices. We rigorously prove the convergence of SQHD for convex and smooth objectives. Numerical experiments demonstrate that SQHD also exhibits advantages in non-convex optimization. All these results highlight its potential for quantum-enhanced machine learning.

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arXiv2024

Continuous Non-monotone DR-submodular Maximization with Down-closed Convex Constraint

We investigate the continuous non-monotone DR-submodular maximization problem subject to a down-closed convex solvable constraint. Our first contribution is to construct an example to demonstrate that (first-order) stationary points can have arbitrarily bad approximation ratios, and they are usually on the boundary of the feasible domain. These findings are in contrast with the monotone case where any stationary point yields a $1/2$-approximation (Hassani et al. (2017)). Moreover, this example offers insights on how to design improved algorithms by avoiding bad stationary points, such as the restricted continuous local search algorithm (Chekuri et al. (2014)) and the aided measured continuous greedy (Buchbinder and Feldman (2019)). However, the analyses in the last two algorithms only work for the discrete domain because both need to invoke the inequality that the multilinear extension of any submodular set function is bounded from below by its Lovasz extension. Our second contribution, therefore, is to remove this restriction and show that both algorithms can be extended to the continuous domain while retaining the same approximation ratios, and hence offering improved approximation ratios over those in Bian et al. (2017a). for the same problem. At last, we also include numerical experiments to demonstrate our algorithms on problems arising from machine learning and artificial intelligence.

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