This paper investigates a presolving method for handling symmetries in mixed-binary programs, based on inequalities computed from so-called Schreier-Sims tables. We show that an iterative application of this method together with merging variables will produce an instance for which the symmetry group is trivial. We then prove that the problem structure can be preserved for certain monotone binary problems, e.g., packing and covering integer programs, maximum stable set (or clique) problems and set covering. The result is then a changed instance of the same type such that existing black-box solvers can directly be applied. We showcase our approach on stable set problems via computational experiments. For a mixed set of graphs, the presolving method significantly outperforms a black-box branch-and-cut method, while being close to the performance of state-of-the-art symmetry handling methods. For Johnson and coding theory graphs, we significantly outperform the stand-alone solver with and without its integrated symmetry handling. A similar speed-up can be observed when running the maximum clique solver CliSAT on the Johnson graphs.
The branch-and-cut paradigm has led to effective solvers for a wide range of mathematical optimization problems. Despite the extensive empirical success and significant theoretical work on pure branching and pure cutting, little is theoretically known about their interaction. In this paper, we study an extended abstract branch-and-cut model and investigate tradeoffs between branching and cutting within the framework. We prove that there always exists a minimal-time tree following a simple root-cut structure, where all cuts are placed before the first branching node, for certain families of settings. For others, although the root-cut structure may be suboptimal and differ substantially from the minimal-time tree, it nevertheless provides a reliable approximation that does not deviate arbitrarily from the actual minimal-time tree. In addition, we identify conditions under which minimal-time trees necessarily contain local cuts, i.e., cuts added after some branching decisions, under a single-cut-per-path restriction. Lastly, simulations indicate that trees restricted to only root cuts significantly outperform our proved worst-case approximation guarantee.
We introduce a novel variant of stochastic load balancing that enables a quantitative trade-off between the practical benefits of non-adaptive policies and their performance limitations. Our model describes a solution in two stages. In the first stage, given only job-size distributions, we reserve a set of at most k machines for each job (a k-reservation). In the second stage, after observing job-size realizations, we assign each job to one of its reserved machines (a consistent assignment). The goal is to minimize the expected makespan. If k=1 , we get the standard stochastic load balancing problem of finding a non-adaptive assignment with minimum expected makespan, and if k is equal to the number of machines, then we obtain an all-powerful omniscient optimum that can tailor the assignment arbitrarily to the job-size realizations. We give a number of results that quantify this trade-off. Most saliently, we show that in the setting of identical machines, a 2-reservation suffices to achieve a constant-factor approximation to the omniscient optimum, establishing a “power-of-two-choices” result for stochastic load balancing. We also show that this no longer holds true in the more challenging setting of related machines. Nonetheless, we give a number of positive algorithmic results for this setting: a true O(log m/loglog m) -approximation; a bicriteria O(1)-approximation by reserving twice as many machines per job relative to an optimal k-reservation; and a 2-reservation whose cost is within a constant factor of the adaptive optimum.
We propose new abstract and unified perspectives on a range of scheduling and graph coloring problems with general min-sum objectives. Specifically, we consider various problems where the objective function is the weighted sum of completion times over groups of entities (jobs, vertices, or edges), thereby generalizing two important objectives in scheduling: makespan and the sum of weighted completion times. As one of our main results, we present a best-possible 𝒪(log g) -competitive algorithm in the non-clairvoyant online setting, where g denotes the size of the largest group. This is the first non-trivial competitive bound for several problems with group completion time objective, and it is an exponential improvement over previous results for non-clairvoyant coflow scheduling. For offline scheduling, we provide elegant yet powerful meta-frameworks that, in a unifying way, yield new or stronger approximation algorithms for our new abstract problems as well as for previously well-studied special cases.
The lattice diameter of a bounded set S ⊂ℝ^d counts the maximal number of lattice points in a segment whose endpoints are lattice points in S. Such a segment is called a lattice diameter segment of S. This basic invariant in the geometry of numbers has several interesting applications. Our first type of contributions are algorithmic: We describe a polynomial-time algorithm that computes lattice diameter segments of polytopes in fixed dimension, with a simpler algorithm for polygons. In contrast we show hardness of computing the lattice diameters of polytopes of variable dimension and for semi-algebraic sets in dimensions three and higher. Our second type of contributions concerns the structure of lattice diameter segments: Generalizing the classical Ehrhart function, we prove that the function that counts distinct lattice diameter segments in dilations of a lattice polytope is eventually a quasi-polynomial in the dilation factor. Finally, we prove a Borsuk-type theorem on the number of parts needed to partition a set of lattice points; such that each part has strictly smaller lattice diameter.
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite relaxation and propose a randomized rounding algorithm to generate feasible solutions from the relaxation. Second, we derive purely multiplicative approximation guarantees for our algorithm. When optimizing for m orthogonal vectors in dimension n, we show that our algorithm achieves a performance ratio of at least max{2π m, 1π (log (2m)+1)} . Our analysis is tight in the sense that we exhibit instances where our algorithm’s performance is at most O(1/log m) . We also show how to compute a tighter constant for finite (n, m) by solving a univariate optimization problem, and this analysis is exact for any n when m=1 .
We propose an O(log n) -approximation algorithm for the bipartiteness ratio of undirected graphs introduced by Trevisan [54], where n is the number of vertices. Our approach extends the cut-matching game framework for sparsest cut to the bipartiteness ratio, and requires only poly log n many single-commodity undirected maximum flow computations. Therefore, with the current fastest undirected max-flow algorithms, it runs in almost linear time. Along the way, we introduce the concept of well-linkedness for skew-symmetric graphs and prove a novel characterization of bipartiteness ratio in terms of well-linkedness in an auxiliary skew-symmetric graph, which may be of independent interest. As an application, we devise an O(mn) -time algorithm for the minimum uncut problem: given a graph whose optimal cut leaves an η fraction of edges uncut, we find a cut that leaves only an O(log n log (1/η )) ·η fraction of edges uncut, where m is the number of edges. Finally, we propose a directed analogue of the bipartiteness ratio, and we give a polynomial-time algorithm that achieves an O(log n) approximation for this measure via a directed Leighton–Rao-style embedding. We also propose an algorithm for the minimum directed uncut problem with a guarantee similar to that for the minimum uncut problem.
The Degree Realization problem requires, given a sequence d of n positive integers, to decide whether there exists problems in which the goal is to compute a realization that optimizes some quality measure. Efficient algorithms are known for the problems of finding a realization with the maximum clique, the maximum independent set, or the minimum vertex cover. In this paper, we focus on the Degree Realization with Min Dominating Set problem for which such algorithm was not known. The goal is to find a realization whose minimum dominating set is minimized among all the realizations of the given sequence d. We present polynomial time realization algorithm for this problem. A related problem of interest and importance is characterizing the sequences with a given value of the optimized function. This leads to an efficient computation of the optimized value without providing the realization that achieves that value. This paper provides a succinct characterization of sequences with a dominating set of a given size.
A rational with denominator a power of two is said to be dyadic. A long standing conjecture of Paul Seymour predicts that for an ideal clutter, the dual of the set covering problem always admits an optimal solution where all variables take dyadic values.We present a local version of this conjecture, and motivate it by proving an important special case.
We introduce a stochastic probing problem with correlated items. In our model, which we call Bayesian Probing, the correlations are modeled by an underlying graph G. Each vertex is independently active with a known probability. Each item corresponds to an edge in the graph. Probing an edge has some cost, gives some reward if both endpoints are active, and also reveals the state of its endpoints. Hence a probe induces a Bayesian update on the remaining edges. The goal is to adaptively probe items/edges subject to a knapsack constraint to maximize the expected total reward obtained from the probed edges. Bayesian Probing generalizes stochastic knapsack and stochastic probing by allowing correlations between items. Moreover, it gives a tractable model for the Bayesian Active Search problem, a popular problem considered in the machine learning community. In this paper, we design efficient approximation algorithms for Bayesian Probing: These results give the first efficient approximation algorithms for Bayesian Active Search, for a class of practically-relevant prior distributions.
The odd-red bipartite perfect matching problem asks to find a perfect matching containing an odd number of red edges in a given red-blue edge-colored bipartite graph. While this problem lies in , its polyhedral structure remains elusive—despite renewed attention to achieving better polyhedral understanding, nurtured by recent advances from two complementary angles. Apart from being a special case of bimodular integer programs, whose polyhedral structure is also badly understood, it is related to one of the most notorious open derandomization questions in theoretical computer science: whether there is a deterministic efficient algorithm for the exact bipartite perfect matching problem, which asks to find a perfect matching with exactly k red edges. Recent progress towards deterministic algorithms for this problem crucially relies on a good polyhedral understanding. Motivated by this, Jia, Svensson, and Yuan show that the extension complexity of the exact bipartite perfect matching polytope is exponential in general. Interestingly, their result is true even for the easier odd-red bipartite perfect matching problem. For this problem, they introduce an exponential-size relaxation and leave open whether it is an exact description. Apart from showing that this description is not exact and even hard to separate over, we show, more importantly, that the red-odd bipartite perfect matching polytope exhibits complex facet structure: any exact description needs constraints with large and diverse coefficients. This rules out classical relaxations based on constraints with all coefficients in {0,± 1} , such as the above-mentioned one, and suggests that significant deviations from prior approaches may be needed to obtain an exact description. More generally, we obtain that also polytopes corresponding to bimodular integer programs have complex facet structure.
This paper studies randomized polynomial kernelization for the weighted d-matroid intersection problem. While the problem is known to have a kernel of size O(d^(k - 1)d) where k is the solution size, the existence of a polynomial kernel is not known, except for the cases when either all the given matroids are partition matroids (i.e., the d-dimensional matching problem) or all the given matroids are linearly representable. The main contribution of this paper is to develop a new kernelization technique for handling general matroids. We first show that the weighted d-matroid intersection problem admits a polynomial kernel when one matroid is arbitrary and the other d-1 matroids are partition matroids. Interestingly, the obtained kernel has size Õ(k^d) , which matches the optimal bound (up to logarithmic factors) for the d-dimensional matching problem. This approach can be adapted to the case when d-1 matroids in the input belong to a more general class of matroids, including graphic, cographic, and transversal matroids. We also show that the problem has a kernel of pseudo-polynomial size when given d-1 matroids are laminar. Our technique finds a kernel such that any feasible solution of a given instance can reach a better solution in the kernel, which is sufficiently versatile to allow us to design parameterized streaming algorithms and faster EPTASs for the budgeted version of matroid intersection.
Let D=(V,A) be a digraph whose underlying undirected graph is 2-edge-connected. A subset of arcs J⊆ A is a strengthening if reversing the arcs of J makes D strongly connected. Given a family ℱ of proper subsets of V, we call a strengthening J tight if after flipping J, there is exactly one arc entering U for every U∈ℱ . We give a polynomial-time algorithm to construct a set ℬ consisting of tight strengthenings which forms an integral basis for the linear hull of tight strengthenings. That is, ℬ is a linearly independent subset of tight strengthenings, and every integer vector in the linear hull of tight strengthenings can be written as an integral combination of ℬ . This extends the main result of Abdi, Cornuéjols, Liu, and Silina (IPCO 2025), who gave a non-constructive proof of the existence of such a basis. While the previous proof uses polyhedral theory, our proof is purely combinatorial and yields a polynomial-time algorithm. As an application of our algorithm, we show that parity-constrained tight strongly connected orientations can be solved in deterministic polynomial time. Along the way, we discover appealing connections to the theory of perfect matching lattices.
In the Weighted Triangle-Free 2-Matching problem (WTF2M), we are given an undirected edge-weighted graph. Our goal is to compute a maximum-weight subgraph that is a 2-matching (i.e., no node has degree more than 2) and triangle-free (i.e., it does not contain any cycle with 3 edges). One of the main motivations for this and related problems is their practical and theoretical connection with the Traveling Salesperson Problem and with some 2-connectivity network design problems. WTF2M is not known to be NP-hard and at the same time no polynomial-time algorithm to solve it is known in the general case (polynomial-time algorithms are known only for some special cases). The best-known (folklore) approximation algorithm for this problem simply computes a maximum-weight 2-matching, and then drops the cheapest edge of each triangle: this gives a 2/3 approximation. In this paper we present a PTAS for WTF2M, i.e., a polynomial-time (1-ε)-approximation algorithm for any given constant ε>0. Our result is based on a simple local-search algorithm and a non-trivial analysis.
Given two matroids ℳ_1 and ℳ_2 over the same ground set, the matroid intersection problem is to find the maximum cardinality common independent set. In the weighted version of the problem, the goal is to find a maximum weight common independent set. It has been a matter of interest to find efficient approximation algorithms for this problem in various settings. In many of these models, there is a gap between the best known results for the unweighted and weighted versions. In this work, we address the question of closing this gap. Our main result is a reduction which converts any α-approximate unweighted matroid intersection algorithm into an α(1-ε)-approximate weighted matroid intersection algorithm, while increasing the runtime of the algorithm by a log W factor, where W is the aspect ratio. Our framework is versatile and translates to settings such as streaming and one-way communication complexity where matroid intersection is well-studied. As a by-product of our techniques, we derive new results for weighted matroid intersection in these models.
The circuit diameter of a polyhedron is the maximum length (number of steps) of a shortest circuit walk between any two vertices of the polyhedron. Introduced by Borgwardt, Finhold and Hemmecke (SIDMA 2015), it is a relaxation of the combinatorial diameter of a polyhedron. These two notions of diameter lower bound the number of iterations taken by circuit augmentation algorithms and the simplex method respectively for solving linear programs. Recently, an analogous lower bound for path-following interior point methods was introduced by Allamigeon, Dadush, Loho, Natura and Végh (SICOMP 2025). Termed straight line complexity, it refers to the minimum number of pieces of any piecewise linear curve that traverses a specified neighborhood of the central path. We study the relationship between circuit diameter and straight line complexity. For a polyhedron P:={x∈ℝ^n: Ax = b, x≥0} , we show that its circuit diameter is up to a poly(n) factor upper bounded by the straight line complexity of linear programs defined over P. This yields a strongly polynomial circuit diameter bound for polyhedra with at most 2 variables per inequality. We also give a circuit augmentation algorithm with matching iteration complexity.
We present a new and faster algorithm for the 4-block integer linear programming problem, overcoming the long-standing runtime barrier faced by previous algorithms that rely on Graver complexity or proximity bounds. The 4-block integer linear programming problem asks to compute min{c_0^⊤ x_0+c_1^⊤ x_1+…+c_n^⊤ x_n | Ax_0+Bx_1+…+Bx_n=b_0, Cx_0+Dx_i=b_i ∀ i∈[n], (x_0,x_1,…,x_n)∈ℤ_≥0^(1+n)k} for some k× k matrices A,B,C,D with coefficients bounded by in absolute value. Our algorithm runs in time f(k,)· n^k+𝒪(1), improving upon the previous best running time of f(k,)· n^k^2+𝒪(1) [Oertel, Paat, and Weismantel (Math. Prog. 2024), Chen, Koutecký, Xu, and Shi (ESA 2020)]. Further, we give the first algorithm that can handle large coefficients in A, B and C, that is, it has a running time that depends only polynomially on the encoding length of these coefficients. We obtain these results by extending the n-fold integer linear programming algorithm of Cslovjecsek, Koutecký, Lassota, Pilipczuk, and Polak (SODA 2024) to incorporate additional global variables x_0. The central technical result is showing that the exhaustive use of the vector rearrangement lemma of Cslovjecsek, Eisenbrand, Pilipczuk, Venzin, and Weismantel (ESA 2021) can be made affine by carefully guessing both the residue of the global variables modulo a large modulus and a face in a suitable hyperplane arrangement among a sufficiently small number of candidates. This facilitates a dynamic high-multiplicy encoding of a faithfully decomposed n-fold ILP with bounded right-hand sides, which we can solve efficiently for each such guess.
Finding a smallest subgraph that is k-edge-connected, or augmenting a k-edge-connected graph with a smallest subset of given candidate edges to become (k+1)-edge-connected, are among the most fundamental Network Design problems. They are both APX-hard in general graphs. However, this hardness does not carry over to the planar setting, which is not well understood, except for very small values of k. One main obstacle in using standard decomposition techniques for planar graphs, like Baker's technique and extensions thereof, is that connectivity requirements are global (rather than local) properties that are not captured by existing frameworks. We present a novel, and arguably clean, decomposition technique for such classical connectivity problems on planar graphs. This technique immediately implies PTASs for the problems of finding a smallest k-edge-connected or k-vertex-connected spanning subgraph of a planar graph for arbitrary k. By leveraging structural results for minimally k-edge-connected graphs, we further obtain a PTAS for planar k-connectivity augmentation for any constant k. We complement this with an NP-hardness result, showing that our results are essentially optimal.
Benders decomposition (BD) is a framework for solving optimization problems by removing some variables and modeling their contribution to the original problem via so-called Benders cuts. While many advanced optimization techniques can be applied in a BD framework, one central technique has not been applied systematically in BD: symmetry handling. The main reason for this is that Benders cuts are not known explicitly but only generated via a separation oracle. In this work, we close this gap by providing a framework of symmetry detection within the BD framework. To this end, we introduce a tailored family of graphs that capture the symmetry information of both the Benders master problem and the Benders oracles. Once symmetries of these graphs are known, which can be found by established techniques, classical symmetry handling approaches become available to accelerate BD. We complement these approaches by devising techniques for the separation and aggregation of symmetric Benders cuts by means of tailored separation routines and extended formulations. Both substantially reduce the number of executions of the separation oracles. In a numerical study, we show the effect of both symmetry handling and cut aggregation for bin packing and scheduling problems.
The presence of symmetries is one of the central structural features that make some integer programs challenging for state-of-the-art solvers. In this work, we study the efficacy of Linear Programming (LP) hierarchies in the presence of symmetries. Our main theorem unveils a connection between the algebraic structure of these relaxations and the geometry of the initial integer-empty polytope: We show that under (k+1) -transitive symmetries–a measure of the underlying symmetry in the problem–the corresponding relaxation at level k of the hierarchy is non-empty if and only if the initial polytope intersects all (n-k) -dimensional faces of the hypercube. In particular, the hierarchies of Sherali-Adams, Lovász-Schrijver, and the Lift-and-Project closure are equally effective at detecting integer emptiness. Our result provides a unifying, group-theoretic characterization of the poor performance of LP-based hierarchies, and offers a simple procedure for proving lower bounds on the integrality gaps of symmetric polytopes under these hierarchies.
