Open research questions in Complexity and Algorithms in Graphs
132 unresolved questions extracted from the limitations and future-work sections of 418 Complexity and Algorithms in Graphs papers in our library. Each links back to the study that raised it.
What the literature leaves open
We prove that the integer-threshold decision problem for weak partition connectivity in hedgegraphs is NP-complete, answering an open question about its computational complexity.
The Complexity of Weak Partition Connectivity in Hedgegraphs · 2026A long-standing open question in query complexity asks whether there is a total Boolean function f with R(f) << C(f), where R(f) and C(f) denote its bounded-error randomized query complexity and certificate complexity, respectively.
Randomized query complexity can beat certificate complexity · 2026Moreover, the (precise) complexity of MMS existence in additive and $k$-additive settings was posed as an open question.
On the Hardness of Maximin Share Allocations · 2026We thereby almost answer an open question by Lokshtanov, Misra, Mukherjee, Panolan, Philip and Saurabh (SODA 2020) who asked for a deterministic 2-approximation in polynomial time.
A deterministic $(2 + \varepsilon)$-approximation for directed feedback vertex sets in tournaments · 2026Whether this problem is fixed-parameter tractable (FPT) in $d$ has remained a central open problem.
High-Multiplicity Bin Packing is FPT · 2026Recently, there have been several works on sparse reliable spanners in various settings, but so far, the weight of such spanners has not been analyzed at all.
Further research can be done to improve the solving of QUBO problems. The application of the proposed mathematical programs to other related problems can be explored. The development of more efficient algorithms for solving the Graph Burning Problem (GBP) can be investigated.
The need for compact mathematical formulations of the Graph Burning Problem. The lack of simple and efficient mathematical programs for the GBP.
Investigating the application of the SR algorithm to other types of graphs, such as directed or weighted graphs. Developing more efficient algorithms for computing maximum quasi-cliques in large graphs. Exploring the use of the SR algorithm in other fields, such as social network analysis or complex network research.
Split-and-Reduce: A New Exact Algorithm with Tight Multi-Order Intersection Bounds for the Quasi-Clique Enumeration Problem · 2026 · DOIThe lack of efficient algorithms for the quasi-clique enumeration problem. The inability of existing methods to handle large-scale graphs. The need for tighter upper bounds on the maximum quasi-clique size.
Split-and-Reduce: A New Exact Algorithm with Tight Multi-Order Intersection Bounds for the Quasi-Clique Enumeration Problem · 2026 · DOIPrior tolerant testers required structural assumptions such as expansion or clusterability. The paper fills this gap by presenting a tolerant tester without such assumptions.
Tolerant Testing for Unique Games · 2026To extend the algorithm to more general models of parallel computation. To improve the algorithm's performance in practice. To apply the algorithm to other problems in distributed computing.
The Task Completion Problem and its Application to Crash-Resilient Computation · 2026Prior work has addressed the Task Completion problem, but with suboptimal round complexity. There is a need for an algorithm with optimal round complexity.
The Task Completion Problem and its Application to Crash-Resilient Computation · 2026The high computational difficulty of the k-defensive domination problem. The need for efficient algorithms to solve large-scale instances of the problem. The complexity of generating Benders cuts.
A Benders Decomposition Approach for the k-Defensive Domination Problem · 2026The lack of efficient algorithms for solving the problem in multipartite graphs. The lack of investigation into the decision, counting, and enumeration variants of the problem.
Complexity of Finding and Enumerating Interconnection Trees · 2026The fixed-bucket estimate loses a factor of 2^ℓ when applied to all buckets. The optimization is limited to the sparse large-load regime. The dense two-sided theorem is not improved by this work.
A Note on Second-Order Expected Maximum-Load Bounds for Binary Linear Hashing · 2026It remains an interesting open problem to scale algorithms to massive graphs. An approximate version of the dynamic program could be solvable in linear time.
An Approximation Algorithm for Graph Label Selection · 2026Prior work relies on resource augmentation or heuristics without provable guarantees. There is a need for an approximation algorithm for graph label selection on general graphs without resource augmentation.
An Approximation Algorithm for Graph Label Selection · 2026Future research could aim to improve the approximation ratio for MAX DI-CUT further. Future research could aim to provide a rigorous proof of the improved upper bound for MAX DI-CUT.
The paper identifies a gap in the approximation ratios for MAX CUT and MAX DI-CUT. The paper identifies a gap in the approximation ratios for MAX DI-CUT and MAX 2-AND.
Future research could focus on improving the derived bounds. The authors suggest exploring the application of semidefinite programming to other graph problems.
Semidefinite Programming Bounds on Fractional Cut-Cover and Maximum 2-SAT for Highly Regular Graphs · 2026 · DOIThe paper identifies a gap in the existing literature on semidefinite programming bounds for highly regular graphs. There is a lack of research on deriving bounds for max 2-SAT using semidefinite programming.
Semidefinite Programming Bounds on Fractional Cut-Cover and Maximum 2-SAT for Highly Regular Graphs · 2026 · DOIThe exact bound on the list replicability number of large-margin halfspaces was unknown. The relationship between list replicability and other notions of stability in learning theory was not well understood.
No comparative performance evaluation exists between parallel exact algorithms (e.g., meet-in-the-middle variants) and parallel metaheuristics for Subset Sum; [1] improves sequential meet-in-the-middle but does not parallelize it, while [4] studies set-based PSO convergence for discrete problems but not specifically for Subset Sum.
No parallel algorithm for subset sum is presented in the literature reviewed; [1] addresses exact algorithms for subset balancing generalizations but does not discuss parallelization strategies, while [12] demonstrates parallelization via clustering for TSP but does not extend this approach to subset sum problems.
Most-cited papers in Complexity and Algorithms in Graphs
- Branch-and-Price: Column Generation for Solving Huge Integer Programs · Operations Research · 1998 · 1,533 citations
- Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning · Operations Research · 1989 · 945 citations
- The Traveling-Salesman Problem and Minimum Spanning Trees · Operations Research · 1970 · 901 citations
- Increasing tree search efficiency for constraint satisfaction problems · Artificial Intelligence · 1980 · 688 citations
- Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph · Operations Research · 1968 · 617 citations
- An analysis of alpha-beta pruning · Artificial Intelligence · 1975 · 608 citations
- Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria · Operations Research · 1981 · 600 citations
- Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning · Operations Research · 1991 · 587 citations
- Modeling and Solving the Train Timetabling Problem · Operations Research · 2002 · 468 citations
- Sequencing a One State-Variable Machine: A Solvable Case of the Traveling Salesman Problem · Operations Research · 1964 · 423 citations
Most recent work
- The Hardest Problem · Scientific American · 2026
- Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers · 2026
- Separator Theorem for Minor-Free Graphs in Linear Time · 2026
- 3-Query RLDCs Are Strictly Stronger Than 3-Query LDCs · 2026
- Lower Bounds against the Ideal Proof System in Finite Fields · 2026
- Graph Burning: An Overview of Compact Mathematical Programs · Mathematics · 2026
- The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs · SIAM Journal on Discrete Mathematics · 2026
- Split-and-Reduce: A New Exact Algorithm with Tight Multi-Order Intersection Bounds for the Quasi-Clique Enumeration Problem · Mathematics · 2026
- Maximal Biclique Enumeration with Improved Worst-Case Time Complexity Guarantee: A Partition-Oriented Strategy · Proceedings of the ACM on Management of Data · 2026
- The approximation algorithm and fast algorithm for constrained or-submodular maximization problem · Journal of Combinatorial Optimization · 2026
Find a gap in your own Complexity and Algorithms in Graphs sub-topic
This page shows what the Complexity and Algorithms in Graphs literature already flags as unresolved. To narrow it to your specific question, search the Research Gap Finder: the search is free with a free account and lists the papers closest to your topic first. Unlocking that topic (50 credits, charged once) fills the comparison table from our 4.5M-paper local library and writes the gaps from its rows.
Open the Research Gap Finder →