tight instance construction

This paper investigates the parameterized complexity of Integer Linear Programming (ILP) with lower and upper bounds, parameterized by the “distance to generalized matching”—the minimum number of modifications required to transform the constraint matrix into one where each column has ℓ₁-norm at most two (encompassing polynomial-time solvable matching and flow problems). The authors introduce two novel structural parameters: a *variable backdoor* (minimum column deletions to achieve the generalized matching structure) and a *constraint backdoor* (minimum row deletions). They present the first fixed-parameter tractable (FPT) algorithm parameterized by variable backdoor size (p). In contrast, they prove W[1]-hardness parameterized by constraint backdoor size (h) and devise a randomized XP algorithm for unary-encoded instances. Key technical innovations include a variant of lattice convexity, Graver basis-enhanced local search, and a pseudo-polynomial reduction to Exact Matching—enabling both tight complexity classification and significant algorithmic advances.

This work addresses compact integer linear programming (ILP) modeling for parameterized problems—specifically, polynomial-time reduction of an instance ((I, k)) to an ILP with only ( ext{poly}(k)) constraints. For problems admitting no polynomial kernel, establishing theoretical connections between WK[1]-hardness and compact ILP modeling remains open. Method: We introduce a novel preprocessing framework based on fast witness verification protocols, circumventing classical kernelization limitations and enabling a new compression paradigm. Integrating data structure optimizations with protocol design, we construct explicit ILP and mixed-integer linear programming (MILP) formulations for classic problems—including (r)-Way Cut, Steiner Tree, and Weighted Vertex Cover—whose constraint counts depend solely on the parameter (k). Results: Our models achieve provably compact formulations with ( ext{poly}(k)) constraints, and empirical evaluation demonstrates substantial improvements in solver efficiency. The approach provides a theoretically grounded yet practically effective modeling pathway for computationally hard parameterized problems.

This work investigates the tractability and computational hardness of Promise Constraint Satisfaction Problems (PCSPs) over Boolean domains. By introducing Fourier-analytic techniques into the PCSP framework—specifically leveraging influence measures of Boolean functions in conjunction with random 2-to-1 minors and sharp threshold theory—the study uncovers two universal mechanisms that govern whether a given problem is efficiently solvable or computationally intractable: the preservation of coordinate influences and the existence of sharp thresholds. This approach extends the prevailing paradigm for ordered PCSPs and, for the first time, establishes a clear dichotomy of tractability within broader classes of Boolean functions, including unate functions and polynomial threshold functions, thereby yielding new complexity-theoretic characterizations.

This paper investigates the fine-grained parameterized complexity of identification problems on graphs and set systems, focusing on Locating-Dominating Set and Test Cover. Under the Exponential Time Hypothesis (ETH), we establish the first tight lower bounds parameterized by solution size $k$: a $2^{Omega(k^2)}$ time lower bound and a $2^{Omega(k)}$ kernel lower bound for Locating-Dominating Set; and—remarkably—a $2^{Omega(k^2)}$ double-exponential lower bound for Test Cover, representing a rare breakthrough in its complexity landscape. Complementing these hardness results, we design matching upper-bound algorithms, thereby resolving the parameterized complexity of identification problems jointly parameterized by treewidth and solution size $k$. Our techniques include problem-specific reductions, tailored auxiliary graph constructions, and tight parameterized analysis.

This work addresses automatic runtime and variable-size bound analysis for integer programs, focusing on the decidable subclass of periodic rational-solvable loops (PRS-loops). The proposed method introduces a modular analysis framework: it first derives local bounds for PRS-loops, then lifts them to global bounds via program transformation and inductive reasoning. Crucially, it extends the decidability of PRS-loop analysis to arbitrary integer programs by designing a synergistic synthesis mechanism combining abstract interpretation with rational linear algebra. The approach is fully automated in the tool KoAT, supporting precise derivation of polynomial and exponential complexity bounds as well as variable growth bounds. Experimental evaluation demonstrates effectiveness on diverse nontrivial integer programs, significantly improving the completeness, precision, and practicality of automated complexity analysis.

This work addresses long-standing open problems in extremal set theory, such as Chvátal’s conjecture, by introducing a novel paradigm for customized search space partitioning based on solution construction strategies, replacing conventional domain-agnostic lookahead-based methods. By integrating this approach with a proof-generating exact mixed-integer linear programming (MILP) solver, the proposed framework substantially enhances search efficiency. Empirical evaluation demonstrates successful verification of the largest finite instance of Chvátal’s conjecture to date, marking significant progress toward resolving this fundamental problem in combinatorics.

This study addresses the long-standing open problem of the integrality gap of the Bidirected Cut Relaxation (BCR) for the Steiner tree problem. By generalizing the dual-fitting primal-dual growth process into a broader “moat-growing algorithm” framework and combining linear programming duality with a tightness analysis, the authors significantly improve the upper bound on the integrality gap in the general case to 1.898. Furthermore, they prove that any moat-growing algorithm cannot achieve a performance ratio better than 12/7 in the special case where the minimum spanning tree over terminals is itself an optimal Steiner tree. This limitation is shown to be intrinsically linked to the structure of hypergraph relaxations, revealing a deep connection between algorithmic barriers and extended formulations.

This work addresses the limited tightness of convex relaxations in mixed-integer nonlinear programming (MINLP) by proposing a computational geometry–based polyhedral relaxation method. It employs a convexification strategy that selects points to iteratively approximate the simultaneous convex hull of factorable function graphs and introduces novel explicit inequalities to strengthen factorable relaxations. Theoretically, it proves that for multilinear functions over axis-aligned domains, the simultaneous convex hull is uniquely determined by its corner points. Furthermore, the approach integrates voxelization with the QuickHull algorithm to efficiently approximate feasible regions. Computational experiments demonstrate that the method reduces the dual gap by 20–25% on average for random polynomial problems and outperforms existing techniques on approximately 30% of MINLPLib instances, with over 10% of cases achieving more than a 50% gap reduction.

This study investigates the long-standing conjecture that the integrality gap of the subtour relaxation for the metric Traveling Salesman Problem (TSP) is 4/3. Building upon the framework of Benoit and Boyd (2008), we systematically analyze the vertex structure of the subtour polytope by integrating polyhedral enumeration, linear programming, and symmetry-based pruning techniques. We correct the known vertex lists for instances with $n = 11$ and $n = 12$, and extend the enumeration to general instances up to $n = 14$ and half-integral instances up to $n = 17$, providing the first complete verification of all vertices in these ranges. Our results confirm that the 4/3 integrality gap conjecture holds for all general instances with $n \leq 14$ and all half-integral instances with $n \leq 17$, offering the strongest empirical support to date for this conjecture.