local-search ptas

This work addresses five classic stochastic combinatorial optimization problems—Free-Order Prophet Inequalities, Pandora’s Box with Commitment, and Adaptive/Non-adaptive ProbeMax—and presents the first unified, efficient polynomial-time approximation scheme (EPTAS): for any ε > 0, it achieves a (1−ε)-approximation in t(ε)·poly(|I|) time. Departing from prior approaches constrained by single-dimensional APX-hardness, the authors introduce the multi-dimensional Santa Claus problem as a novel reduction target. Their method features problem-specific reductions, multi-dimensional resource modeling, structured dynamic programming, and error-controlled discretization. The result yields the first practical EPTAS for all five problems, with the ε-dependent factor fully decoupled from input size—marking a substantial improvement over previous existential, non-scalable, and computationally inefficient PTASes. Notably, this is the first efficient approximation scheme for the non-adaptive setting, closing a long-standing gap in the literature.

This paper investigates the computational complexity of finding two distinct local optima in unweighted local search problems. For classical combinatorial optimization problems—including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut—it establishes that computing any two distinct local optima under standard neighborhood structures is NP-hard, contrasting sharply with the fact that a single local optimum can be found in polynomial time (i.e., these problems lie in PLS). The work provides the first systematic characterization of the inherent hardness of multi-solution local search, thereby extending the boundaries of PLS complexity theory. Through carefully constructed polynomial-time reductions and combinatorial structural analysis, it proves NP-hardness for the two-solution variants. Additionally, it identifies several special cases—such as bounded-degree graphs or structured constraint families—in which multiple local optima can be computed efficiently, offering a new paradigm for studying solution multiplicity in local search.

This work addresses parameterized approximation algorithms for Vertex Cover and 3-Hitting Set. Methodologically, it introduces a novel randomized branching paradigm grounded in an equivalence between the algorithm’s recursive structure and a binary stochastic process. Leveraging a type-theoretic adaptation of Sanov’s theorem, the framework performs large-deviation analysis on bivariate recurrence relations, yielding an analytically tractable master theorem for asymptotic running time. Contribution-wise, this is the first unified theoretical framework providing rigorous approximation-ratio–dependent guarantees across multiple approximation factors. It substantially improves worst-case time complexity over prior deterministic branching approaches and overcomes fundamental analytical limitations inherent in traditional branching analysis. The framework establishes a general methodology for characterizing the asymptotic performance of parameterized approximation algorithms, bridging stochastic analysis and combinatorial optimization.

This paper investigates performance optimization of Local Computation Algorithms (LCAs) on random graphs under average-case analysis. Addressing the Erdős–Rényi and preferential attachment models, it pioneers the extension of the LCA framework to the average-case setting, designing sublinear-query local access schemes for combinatorial structures—including *k*-spanners and maximum independent sets. The main contributions are threefold: (1) It demonstrates that structural properties of random graphs enable circumvention of worst-case stretch–size trade-offs, yielding improved parameter balances; (2) It introduces a novel “joint generation” paradigm, simultaneously constructing the random graph instance and its target combinatorial structure; (3) It constructs the first efficient LCA that locally generates a maximum independent set on ER graphs with query complexity significantly below the worst-case bound, and achieves optimal stretch–size trade-offs for *k*-spanner access on both models.

This work proposes a novel local computation algorithm (LCA) for the set cover problem that significantly reduces query complexity while preserving approximation quality. The key innovation lies in an aggressive input sparsification strategy combined with a “backtrack-and-update” mechanism, which enables the algorithm to dynamically revise decisions made in earlier recursive calls. This enhances solution concentration and curtails redundant computation. Theoretical analysis demonstrates that the proposed method lowers the query complexity from Δ^{O(log Δ)} to f^{O(log Δ)}; moreover, when f = polylog Δ, the complexity is further improved to Δ^{O(log log Δ)}, substantially outperforming existing approaches.

This work addresses the limitations of traditional complexity frameworks—such as PLS—in capturing the core computational challenges inherent in designing efficient pivoting rules for local search. The authors propose a novel framework that requires algorithms to output not only a locally optimal solution but also the complete improvement path leading to it. By integrating parameterized complexity theory with the entire trajectory of local search and focusing on improvement chains rather than individual steps, this approach more accurately models the computational hardness of pivoting rules. Using a new form of reduction, the study analyzes the fixed-parameter tractability of canonical problems—including Subset Weight Optimization and Weighted Circuit—under c-swap and flip neighborhoods. It shows that these problems are efficiently solvable when parameterized by the number of weights, yet become intractable when parameterized by the distance to an optimal solution.

Under the Gap-ETH assumption, this work establishes the conditional optimality of shifting-based PTAS algorithms for a range of d-dimensional unit ball graph problems—including Maximum Independent Set and Maximum Induced Forest—as well as the Unit Ball Piercing problem, all of which admit running times of $n^{O(1/\varepsilon^{d-1})}$. This result extends, for the first time, the known optimality of shifting techniques from two dimensions to arbitrary constant dimensions. To achieve this, the authors develop a unified maximization framework for geometric constraint satisfaction problems (CSPs), integrating De Berg’s cube wiring theorem with the reduction methodology of Marx and Sidiropoulos. This framework provides a common foundation for proving conditional hardness results across high-dimensional geometric optimization problems.

This work addresses the challenge of efficiently extending distributed constant-round approximation algorithms for planar graphs to broader classes of “locally well-behaved” graphs, such as bounded-genus graphs and those with finite asymptotic dimension, specifically for cuttable minimization problems like Minimum Dominating Set. The authors propose a general meta-theorem that establishes the first unified framework enabling such systematic extensions in the LOCAL model. By integrating techniques from graph embeddings, asymptotic dimension theory, and local solution sparsity analysis, their approach significantly improves approximation guarantees: on genus-$g$ graphs, the approximation ratio for Minimum Dominating Set is reduced from $24g + O(1)$ to $34 + \varepsilon$, and the method further extends to variants such as $k$-tuple dominating set, outperforming the previous best bound of $91 + \varepsilon$ on orientable surfaces.