hop-based topology traversal

This paper investigates the theoretical capabilities and efficiency of greedy algorithms for constructing shortcut sets and hopsets in directed weighted graphs. We propose a deterministic greedy strategy that leverages path covering and transitive closure techniques to build exact β-hopsets—without relying on random sampling. Our main contributions are threefold: (1) We construct exact hopsets of size $ ilde{O}(n)$ with hopbound $O(n^{1/3})$, matching the optimal size upper bound established at SODA’22; (2) We design a deterministic algorithm achieving time complexity $O(mn^{2/3})$; (3) Under certain conditions, we attain existence-optimal exact hopsets—providing the first systematic evidence that greedy methods possess both theoretical promise and practical utility for graph sparsification and path compression.

This paper addresses the high computational complexity of parallel algorithms for single-source reachability and shortest paths on directed graphs. We propose a generic black-box framework that applies a shallow graph transformation to convert any input graph into an equivalent, structurally simpler, and shallower graph—enabling existing shortcut- and hopset-based parallel algorithms to run directly and efficiently on the transformed instance. Crucially, our framework decouples shortcut and hopset construction while weakening structural assumptions previously required on the original graph. Leveraging near-linear-work parallel graph primitives, our approach significantly simplifies the design and analysis of multiple classical algorithms, enhancing both theoretical interpretability and practical implementability. The framework is model-agnostic, supporting diverse parallel computing models including PRAM and MPC.

This paper addresses the Vertex Cover problem by proposing the first automated framework for generating branching rules and enhancing the Measure & Conquer technique to enable systematic optimization of randomized branching algorithms. Methodologically, it integrates local structural analysis, adaptive randomized branching strategies, and structural properties of bounded-degree graphs (Δ ≤ 6), significantly improving both branching efficiency and flexibility. The main contributions are: (1) the first automated derivation mechanism for branching rules; (2) the current best randomized time bounds for cubic graphs—O*(1.07625ⁿ) in terms of input size n and O*(1.13132ᵏ) in terms of solution size k; and (3) new state-of-the-art randomized time complexities for Vertex Cover on both bounded-degree graphs (Δ ≤ 6) and general graphs, advancing the frontier of algorithmic performance in this domain.

This work addresses the long-standing challenge of achieving subquadratic query complexity for fundamental problems such as reachability in directed graphs under the cut-query model. Focusing on directed acyclic graphs (DAGs), the authors combine divide-and-conquer techniques with structural properties of DAGs to design the first algorithm with nearly linear query complexity. Specifically, single-source reachability and topological ordering can be determined using $O(n \log^3 n)$ cut queries. The approach is further extended to detect cycles in general directed graphs within the same asymptotic query bound. This result breaks a longstanding efficiency barrier in the cut-query model for directed graphs, marking a significant theoretical advance in the field.

This paper studies the Weighted Expansion Search problem on graphs: a searcher starts from a designated vertex and iteratively expands along edges to unvisited vertices, with edge lengths representing traversal time; the objective is to minimize the weighted sum of vertex weights multiplied by their first-visit times. As an NP-hard problem, it had long been stuck at an 8-approximation barrier. We break this barrier via a unified framework integrating greedy construction, randomized rounding, Euclidean geometric partitioning, and dynamic programming, complemented by hardness analysis under exponential-time hypotheses. Our contributions are the first: (i) a $(2e+varepsilon)$-approximation algorithm for general graphs; (ii) a $2e$-approximation algorithm for unit-weight graphs; and (iii) a PTAS for Euclidean graphs (for any $varepsilon > 0$). These results substantially improve approximation ratios and provide the strongest known theoretical guarantees for all three graph classes.

This work proposes the first parallel algorithm for single-source reachability and shortest paths on non-sparse directed graphs that achieves near-linear work (Õ(m)) and depth strictly below √n. By integrating graph decomposition, recursive contraction, and parallel search—augmented with refined load balancing and depth-optimized scheduling—the algorithm attains o(√n) depth on graphs with m ≥ n^{1+o(1)} edges. In dense regimes, such as when m = Ω(n²), the depth improves further to n^{0.136} for reachability and n^{0.25+o(1)} for shortest paths. These results substantially advance the state of the art by overcoming longstanding bottlenecks in the depth–work trade-off for these fundamental graph problems.

This work addresses the single-source reachability problem in directed acyclic graphs (DAGs) under the cut-query model. To overcome the high query cost of traditional approaches that require full graph reconstruction, the authors propose a deterministic algorithm that achieves subquadratic query complexity—the first such result for DAGs in this model. By integrating topological ordering with cut queries, the method efficiently infers reachability relations without explicitly constructing the graph structure and naturally extends to single-source shortest paths. The algorithm requires only $O(n\sqrt{n \log n})$ queries, significantly improving upon the previous randomized graph-reconstruction approach with $O(n^2 / \log n)$ query complexity.

This study addresses the maximum cut problem on planar graphs with arbitrary edge weights and establishes tight upper bounds alongside an efficient heuristic algorithm for toroidal graphs. By reformulating the problem as a minimum T-join problem on the dual graph, the authors extend Hadlock’s algorithm—originally limited to unweighted instances—to handle arbitrary weights for the first time. Leveraging this generalized framework, they derive theoretically tight upper bounds for toroidal graphs and propose a novel heuristic grounded in planar graph solutions. Experimental evaluation on the GSet toroidal benchmark suite confirms the optimality of several previously known maximum cut values and yields a new best-known solution for instance #62.

This work addresses the high computational complexity of cut-set computation in multi-path ensemble attribute evaluation by proposing an efficient algorithm and developing a vectorized computing framework based on matrix operations, which reformulates path attribute calculations as parallelizable array operations. For the first time, this approach provides a practical implementation of the formal model for path set attributes, integrating an optimized cut-set algorithm with array-oriented programming languages to substantially improve computational efficiency. Empirical evaluations across network simulations of varying complexity demonstrate that the method yields predictable and acceptable execution times, thereby establishing a practical foundation for large-scale multi-path analysis.

This study addresses the inefficiency in constructing distance structures for directed graphs by proposing faster algorithms for directed (1+ε)-hopsets, deterministic shortcut sets, and source-wise distance preservers. Methodologically, this work presents the first efficient algorithms that match the latest theoretical bounds, extending them to general graphs via DAG projection techniques while optimizing the computational pipeline through structured hopset analysis and accelerated reduction strategies. By significantly improving the construction speed of directed graph distance structures and achieving state-of-the-art trade-offs between size and hop count, this project provides novel theoretical and practical foundations for shortest path problems in directed graphs.