Open research questions in Complexity and Algorithms in Graphs
26 unresolved questions extracted from the limitations and future-work sections of 302 Complexity and Algorithms in Graphs papers in our library. Each links back to the study that raised it.
What the literature leaves open
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.
No parallel algorithm design or performance evaluation framework exists for the Subset Sum problem itself; [1] addresses generalized Subset Balancing but does not discuss parallelization strategies, and [3] provides a generic parallel branch-and-bound template without instantiation to Subset Sum or empirical parallel performance metrics on this specific problem.
We further highlight the importance of the constant-sized finite field regime in IPS by showing that any hard instance in this regime for a sufficiently strong proof system translates into a hard instance against AC0[p]-Frege, whose lower bounds remain a longstanding open problem.
This work establishes lower bounds against fragments of IPS over constant-sized finite fields, resolving an open problem left by a series of prior works beginning with Forbes, Shpilka, Tzameret, and Wigderson (Theor.
In this paper, we have presented a novel approach to compute the integrality gap of small atsp instances that enabled the computation of the exact integrality gap for halfinteger vertices for n = 10, 11, 12. In our algorithm, we first generate pairs of edge-disjoint cycle covers through a lexicographically ordered encoding that filters out several isomorphic instances, and later, we efficiently check a few properties that characterize those cycle covers as a vertex of asep. In future works, we plan to apply our approach to larger values of n and to k-tuple of cycle covers, k 3, to study different classes of fractional vertices, and to adapt the described approach to the symmetric tsp. ≥ 1 3The cloven traveling salesman: cycle covers and the integrality gap of… Appendix A: Algorithms In this appendix, we report the pseudocode of the algorithm described in the paper. The corresponding implementation in Python is available at. Algorithm 1 Cover encoding Algorithm 2 Standard cover-set encoding 1 3A. Sosso et al. Algorithm 3 Circuit extremality algorithm Appendix B: Canonical form of a cover-set encoding The standard form of a cover-set encoding is not unique. One may define the canonical form of a cover-set encoding the standard form that achieves the minimum in a lexicographic order among all possible standard forms (the comparison between elements of the standard forms is again done lexicographically). By construction, 1 3The cloven traveling salesman: cycle covers and the integrality gap of… Fig. 8 Isomorphic vertices with different cover-set canonical encodings. Note that the isomorphism sending the first one into the second one is represented by σ:= (015)(234) this canonical form is therefore unique for each set-cover encoding. This construction presents two issues. First, producing canonical instances indeed requires exploring all their possible standard translations to check whether the minimum has been reached, obviously an extremely demanding operation. Notice also how it is, in fact, equivalent to generating all standard forms and then checking for their canonicity afterward. Second, counterexamples, as shown in Fig. 8, display two different cover sets (even with different partitions) in canonical form that represent two isomorphic vertices. The canonical form is therefore not strong enough to identify the isomorphism class of a cover-set. It is therefore equivalent and better for us to generate all standard encodings, since it leaves us the choice of whether to check afterward for their canonicity or directly for their isomorphism class with faster algorithms. In particular, it is computationally more efficient to just check if a generated instance is isomorphic to an instance we already encountered. Appendix C: Omitted Proofs Proof of Proposition 4 We already remarked how ←→A⊆←→A′, and we now cone2] struct explicitly the elements in can only be involved in the linkage of one of its extremities to arcs adjacent (with respect to A) to the other one, that is ←→A′ \ ←→A. We observe that the element [e1| e1 e2] [e1| ←→ A′ e2 ∀ = and adjA (e2) e2 ∈ (Remark 2) and adjA (e2) ∅ e2 e2] [e1| ←→ A′ e1 ∀ L2 e1 ∈, the two equivalence classes adjA (e1). A A Since adjA (e2) L1 and L2 alence relation (eventually non-distinct) class(es): ⊆ having some A′-linked representatives thus merge under the new equiv-. This means that the four ∼A′-equivalence ∼A′. Similarly happens for L2 and L1 ∼A-equivalence classes merge into the A 1 3A. Sosso et al.̸ L1 A is equivalent to L1 = L2); in this case the only circuit L when either L1 = L2, L1 = L1 L2 A or L2 = L2 A ′ is its ∈ L otherwise; the dualism between the two circuits L′ becoming duals by construc- L and e2 ∈ since it would imply the satisfaction of the case (C) LA gets ←→A′ \ ←→A, no other class of ′ =, therefore L ∼A′. For the last part, just note that L1, L1 A i. ii.
The cloven traveling salesman: cycle covers and the integrality gap of small ATSP instances · 2026 · DOIGenetic algorithms [5] and heuristic-based parallelization [12] show promise for combinatorial problems, but no work applies genetic algorithms or population-based parallel methods to the subset sum problem specifically, leaving open whether such approaches can compete with exact parallel methods.
An integer linear programming formulation and genetic algorithm for the maximum set splitting problem · 2012 · DOIComparative performance evaluation of parallel metaheuristic approaches (genetic algorithms, particle swarm optimization) on Subset Sum is absent; [2] and [4] study these methods on related combinatorial problems (set splitting, knapsack) but [4] reveals convergence failures in set-based PSO, and neither evaluates scalability or parallel efficiency on Subset Sum instances.
An integer linear programming formulation and genetic algorithm for the maximum set splitting problem · 2012 · DOIThe structured constraint taxonomy enabling closed-form infeasibility detection via a scalar ratio φ is not addressed in existing integer programming literature. Prior work on infeasibility and cutting planes does not exploit domain-specific constraint structures in multi-product manufacturing to derive closed-form detection rules.
No prior work integrates automated relaxation coefficient selection with branch-and-bound search in a unified framework for integer programming. Existing decomposition and relaxation methods (Dantzig-Wolfe, Benders, RLT hierarchies) focus on problem reformulation or cut generation but do not systematically optimize relaxation parameters per-node during branch-and-bound exploration for production planning.
On Dantzig-Wolfe Decomposition in Integer Programming and ways to Perform Branching in a Branch-and-Price Algorithm · 2000 · DOINo exploration of hybrid parallel algorithms combining exact methods (branch-and-bound) with heuristic acceleration for Subset Sum; [8] demonstrates hybrid evolutionary methods for scheduling, but Subset Sum lacks a comparable hybrid framework that leverages both lower bounds from dual relaxations ([6]) and parallel metaheuristic search.
Lagrangian relaxation and branch-and-bound are effective for integer programming [11] and set partitioning [3, 4], but their application to subset sum with explicit parallel decomposition strategies—such as partitioning the search space or dual bounds across processors—is not explored in the provided literature.
No study combines dual heuristics with parallel branch-and-bound for Subset Sum; [6] demonstrates dual LP relaxation heuristics for set covering/partitioning problems, but parallelization of this bounding strategy and its interaction with parallel search on Subset Sum remain unexplored.
The theoretical guarantees for δ-monotone bound tightening and LP upper-bound validity under iterative relaxation within branch-and-bound are not established in prior work. Existing branch-and-bound theory specifies convergence conditions but does not characterize how relaxation coefficient monotonicity preserves bound quality across the search tree.
In this paper, we give a general method for designing TLFPT algorithms parameterized by treewidth, solving three open problems posed by Bumpus et al.
Courcelle's Theorem in Truly Linear FPT · 2026This resolves open questions raised by Kazachkov, Le Bodic, and Sankaranarayanan on minimal-size trees under harmonically-worsening cuts, and generalizes their results to asymmetric branching and to all cut schedules in the model with this logarithmic limit.
Sharp Logarithmic Thresholds for Cut Schedules in an Abstract Branch-and-Cut Model · 2026Meet-in-the-middle is the classical baseline for subset sum, and [1] beats it for subset balancing, but no work addresses how to parallelize meet-in-the-middle itself or whether parallel variants can maintain its theoretical advantages on modern multi-core architectures.
Strong coresets preserve the cost of every candidate solution but require linear time to construct; weak coresets admit sublinear construction, in fact by uniform sampling, but only preserve near-optimal solutions, which is insufficient when the solution is constrained.
Optimal Stable Coresets for Geometric Median via Uniform Sampling · 2026We consider the problem of testing if an unknown low-degree polynomial p over ℝn is sparse versus far from sparse, given access to noisy evaluations of the polynomial p at randomly chosen points.
Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem.
A primary open question in the study of non-redundancy is the identification of which CSP predicates have near-linear NRD.
Super-linear Lower Bounds for CSP Non-Redundancy via Shrinking Instances · 2026K hachian proposed an entirely different solution method whose running time is bounded by a polynomial function of problem size, thereby settling a major open problem in computational complexity theory.
Most-cited papers in Complexity and Algorithms in Graphs
- Optimal Nonbipartite Matching and Its Statistical Applications · The American Statistician · 2011 · 137 citations
- The complexity of linear programming · Statistica Neerlandica · 1981 · 13 citations
- Constructing large peak-pit Condorcet domains · Theory and Decision · 2022 · 11 citations
- Matching Methods for Observational Studies Derived from Large Administrative Databases · Statistical Science · 2020 · 11 citations
- Negative-Weight Single-Source Shortest Paths in Near-Linear Time · Communications of the ACM · 2025 · 4 citations
- Commentary - Methods to find all the edges on any of the shortest paths between two given nodes of a directed acyclic graph · Research Ideas and Outcomes · 2021 · 2 citations
- Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers · 2026 · 1 citations
- Separator Theorem for Minor-Free Graphs in Linear Time · 2026 · 1 citations
- 3-Query RLDCs Are Strictly Stronger Than 3-Query LDCs · 2026 · 1 citations
- Lower Bounds against the Ideal Proof System in Finite Fields · 2026 · 1 citations
Most recent work
- 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
- Separating MAX 2-AND, MAX DI-CUT, and MAX CUT · SIAM Journal on Computing · 2026
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