The reachability diameter (ReachDiam) of a directed graph is the maximum distance over all pairs u,v where v is reachable from u. This notion is present in the definition of shortcut sets, and the name was recently coined in that context by Haeupler, Jiang, and Saranurak [SOSA 2026]. While this is a very natural notion of diameter in directed graphs, and especially DAGs, it is so far not computationally explored. Other definitions of diameter in directed graphs are either trivial (infinite) in graphs that are not strongly connected (e.g., the classical definition) or are non-trivial only in highly restrictive graph classes (e.g., Min-Diameter). We initiate the problem of computing the (approximate) reachability diameter from a fine-grained complexity point of view. Under certain fine-grained assumptions, we prove that there is no algorithm in time 𝒪(n^{ω - ε}) that gives any approximation of ReachDiam in weighted graphs. Similarly, there is no algorithm with better than 2-approximation for unweighted graphs in this time. To supplement this, we provide algorithmic upper bounds that lead to additive approximation of ReachDiam for unweighted graphs. Hence, we establish a strong separation between the weighted and unweighted cases, which makes this type of diameter different in nature than other known notions. Considering the hardness in general weighted graphs, we also study special graph classes and get small constant approximations for DAGs with bounded width or graphs with bounded treewidth. Interestingly, our techniques also lead to exact hopsets with hopbound 2 for bounded treewidth graphs. This and some of our upper bounds for general graphs show technical connections between approximating ReachDiam and computing shortcut sets and hopsets.

Fix a parameter k ∈ ℕ. We give dynamic data structures that for a fully dynamic undirected graph G, updated over time by edge insertions and edge deletions, can answer the following queries: - Long (u,v)-path: Given u,v ∈ V(G), is there a path from u to v of length at least k? - Long (u,v)-detour: Given u,v ∈ V(G), is there a path from u to v of length at least dist_G(u,v)+k? - Even/odd (u,v)-path: Given u,v ∈ V(G), is there a path from u to v of even/odd length? The amortized time of executing an update or answering a query is 2^𝒪(k³) log n + 𝒪(log² n log² log n) in the first two cases, and 𝒪(log² n log² log n) in the last, where n is the number of vertices of G. The first result is in sharp contrast with known conditional lower bounds for reporting paths of length at most k. Specifically, there is no data structure supporting queries about (u,v)-paths of length at most two in time n^o(1) unless the Triangle Conjecture fails. Our main technical contribution is a mechanism of "delayed edge insertion" that works locally on the level of biconnected components.

For a real matrix \(\textbf A \in \mathbb{R}^{d \times n}\) with non-collinear columns, we show that \(n \le O(d^{4} \kappa_\textbf A)\) where \(\kappa_\textbf A\) is the circuit imbalance measure of \(\textbf A\). The circuit imbalance measure \(\kappa\) is a real analogue of \(\Delta\)-modularity for integer matrices, satisfying \(\kappa_\textbf A \le \Delta_\textbf A\) for integer \(\textbf A\). The circuit imbalance measure has numerous applications in the context of linear programming (see Ekkbatani, Natura and Végh (2022) for a survey). Our result generalizes the \(O(d^{4} \Delta_\textbf A)\) bound of Averkov and Schymura (2023) for integer matrices and provides the first polynomial bound holding for all parameter ranges on real matrices.

We provide faster strongly polynomial time algorithms solving maximum flow in structured \(n\)-node \(m\)-arc networks. Our results imply an \(n^{\omega+o(1)}\)-time strongly polynomial time algorithms for computing a maximum bipartite \(b\)-matching where \(\omega\) is the matrix multiplication constant. Additionally, they imply an \(m^{1+o(1)}W\)-time algorithm for solving the problem on graphs with a given tree decomposition of width \(W\).

Abstract. We develop a new “subspace layered least squares" interior point method (IPM) for solving linear programs. Applied to an [Formula: see text]-variable linear program in standard form, the iteration complexity of our IPM is up to an [Formula: see text] factor upper bounded by the straight-line complexity (SLC) of the linear program. This term refers to the minimum number of segments of any piecewise linear curve that traverses the wide neighborhood of the central path, a lower bound on the iteration complexity of any IPM that follows a piecewise linear trajectory along a path induced by a self-concordant barrier. In particular, our algorithm matches the number of iterations of any such IPM up to the same factor [Formula: see text]. As our second contribution, we show that the SLC of any linear program is upper bounded by [Formula: see text], which implies that our IPM’s iteration complexity is at most exponential. This is in contrast to existing iteration complexity bounds that depend on either bit complexity or condition measures; these can be unbounded in the problem dimension. We achieve our upper bound by showing that the central path is well-approximated by a combinatorial proxy we call the max central path, which consists of [Formula: see text] shadow vertex simplex paths. Our upper bound complements the lower bounds of Allamigeon et al. [ SIAM J. Appl. Algebra Geom., 2 (2018), pp. 140–178] and Allamigeon, Gaubert, and Vandame [ No self-concordant barrier interior point method is strongly polynomial, 2022], who constructed linear programs with exponential SLC. Finally, we show that each iteration of our IPM can be implemented in strongly polynomial time. Along the way, we develop a deterministic algorithm that approximates the singular value decomposition of a matrix in strongly polynomial time to high accuracy, which may be of independent interest.

The frame scaling problem is: given vectors , marginals , and precision ɛ > 0, find left and right scalings such that (v1,…,vn) := (Lu1r1,…, Lunrn) simultaneously satisfies and , up to error ɛ. This problem has appeared in a variety of fields throughout linear algebra and computer science. In this work, we give a strongly polynomial algorithm for frame scaling with log(1/ɛ) convergence. This answers a question of Diakonikolas, Tzamos and Kane (STOC 2023), who gave the first strongly polynomial randomized algorithm with poly(1/ɛ) convergence for Forster transformation, the special case . Our algorithm is deterministic, applies for general marginals , and requires O(n3 log(n/ɛ)) iterations as compared to the O(n5d11/ɛ5) iterations of DTK. By lifting the framework of Linial, Samorodnitsky and Wigderson (Combinatorica 2000) for matrix scaling to the frame setting, we are able to simplify both the algorithm and analysis. Our main technical contribution is to generalize the potential analysis of LSW to the frame setting and compute an update step in strongly polynomial time that achieves geometric progress in each iteration. In fact, we can adapt our results to give an improved analysis of strongly polynomial matrix scaling, reducing the O(n5 log(n/ɛ)) iteration bound of LSW to O(n3 log(n/ɛ)). Additionally, we give a bound on the size of approximate scaling solutions, which involves condition measure studied in the linear programming literature, and may be of independent interest.

We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Végh ‍(MOR ’17) and Megiddo ‍(SICOMP ’83), respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) by Allamigeon, Dadush, Loho, Natura, and Végh ‍(FOCS ’22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum ‍(ORL ’04), the same bound applies to any linear program with at most two non-zeros per column or per row. To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices and extreme rays. Our approach is black-box and can be applied to any log-barrier path-following method.

No abstract available.

In this work, we prove new bounds on the additive gap between the value of a random integer program max cTx, Ax ≤ b, x ∈ {0,1}n with m constraints and that of its linear programming relaxation for a wide range of distributions on (A,b,c). Our investigation is motivated by the work of Dey, Dubey, and Molinaro (SODA'21), who gave a framework for relating the size of Branch-and-Bound (B&B) trees to additive integrality gaps. Dyer and Frieze (MOR '89) and Borst et al. (Mathematical Programming '22), respectively, showed that for certain random packing and Gaussian IPs, where the entries of A, c are independently distributed according to either the uniform distribution on [0,1] or the Gaussian distribution N(0,1), the integrality gap is bounded by Om(log2 n/n) with probability at least 1 − 1/n - e−Ωm(1). In this paper, we generalize these results to the cases where the entries of A are uniformly distributed on an integer interval (e.g., entries in {-1,0,1}), and where the columns of A are distributed according to an isotropic logconcave distribution. Second, we substantially improve the success probability to 1 - 1/poly(n), compared to constant probability in prior works (depending on m). Leveraging the connection to Branch-and-Bound, our gap results imply that for these IPs B&B trees have size npoly(m) with high probability (i.e., polynomial for fixed m), which significantly extends the class of IPs for which B&B is known to be polynomial. Our main technical contribution and the key to achieving the above results is a new linear discrepancy theorem for random matrices. Our theorem gives general conditions under which a target vector is equal to or very close to a {0,1} combination of the columns of a random matrix A. Compared to prior results, our theorem handles a much wider range of distributions on A, both continuous and discrete, and achieves success probability exponentially close to 1, as opposed to the constant probability shown in earlier results. Our proof uses a Fourier analytic approach, building on the work of Hoberg and Rothvoss (SODA '19) and Franks and Saks (RSA '20) who studied the discrepancy of random set systems and matrices respectively.

The combinatorial diameter $\operatorname{diam}(P)$ of a polytope $P$ is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an $n$-dimensional polytope $P$ defined by the intersection of $m$ i.i.d.\ half-spaces whose normals are chosen uniformly from the sphere, we show that $\operatorname{diam}(P)$ is $Ω(n m^{\frac{1}{n-1}})$ and $O(n^2 m^{\frac{1}{n-1}} + n^5 4^n)$ with high probability when $m \geq 2^{Ω(n)}$. For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when $m$ is large, where we rely on the $Θ(n^2 m^{\frac{1}{n-1}})$ bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these ``shadows paths'' together over a suitable net using worst-case diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope $P^\circ$, corresponding to a random convex hull, by showing the relation $\operatorname{diam}(P) \geq (n-1)(\operatorname{diam}(P^\circ)-2)$. We then prove that the shortest path between any ``nearly'' antipodal pair vertices of $P^\circ$ has length $Ω(m^{\frac{1}{n-1}})$.

Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound $O(2^{n}n^{15}\log n)$ for an n-variable linear program in standard form. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations. The number of iterations of our algorithm is at most $O(n^{15}\log n)$ times the number of segments of any piecewise linear curve in the wide neighborhood of the central path. In particular, it matches the number of iterations of any path following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the max central path’, a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths. From the existence of a line segment in the wide neighborhood we derive strong implications on the structure of the corresponding segment of the central path. Our algorithm is able to detect this structure from the local geometry at the current iterate, and constructs a step direction that descends along this segment. The bound $O(n^{15}\log n)$ that applies for arbitrarily long line segments is derived from a combinatorial progress measure. Our algorithm falls into the family of layered least squares interior point methods introduced by Vavasis and Ye (Math. Prog. 1996). In contrast to previous layered least squares methods that partition the kernel of the constraint matrix into coordinate subspaces, our method creates layers based on a general subspace providing more flexibility. Our result also implies the same bound on the number of iterations of the trust region interior point method by Lan, Monteiro, and Tsuchiya (SIOPT 2009).

We consider linear programming in the oracle model: mincT x s.t. x ∊ P, where the polyhedron P = {x ∊ ℝn: Ax ≤ b} is given by a separation oracle that returns violated inequalities from the system Ax ≤ b. We present an algorithm that finds exact primal and dual solutions using O(n2 log(n/δ)) oracle calls and O(n4 log(n/δ) + n6 log log(1/δ)) arithmetic operations, where δ is a geometric condition number associated with the system (A, b). These bounds do not depend on the cost vector c. The algorithm works in a black box manner, requiring a subroutine for approximate primal and dual solutions; the above running times are achieved when using the cutting plane method of Jiang, Lee, Song, and Wong (STOC 2020) for this subroutine. Whereas approximate solvers may return primal solutions only, we develop a general framework for extracting dual certificates based on the work of Burrell and Todd (Math. Oper. Res. 1985). Our algorithm works in the real model of computation, and extends results by Grötschel, Lovász, and Schrijver (Prog. Comb. Opt. 1984), and by Frank and Tardos (Combinatorica 1987) on solving LPs in the bit-complexity model. We show that under a natural assumption, simultaneous Diophantine approximation in these results can be avoided.

Motivated by the Matrix Spencer conjecture, we study the problem of finding signed sums of matrices with a small matrix norm. A well-known strategy to obtain these signs is to prove, given matrices A1, …, An ∈ ℝm × m, a Gaussian measure lower bound of 2−O(n) for a scaling of the discrepancy body {x ∈ ℝn: || ∑i=1n xi Ai|| ≤ 1}. We show this is equivalent to covering its polar with 2O(n) translates of the cube 1/n B∞n, and construct such a cover via mirror descent. As applications of our framework, we show:

The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive quantitative bounds on the expected suprema and the degree of continuity of sample paths for many processes. One of the crowning achievements of the theory is Talagrand’s tight alternative characterization of the suprema of Gaussian processes in terms of majorizing measures. The proof of this theorem was difficult, and thus considerable effort was put into the task of developing both shorter and easier to understand proofs. A major reason for this difficulty was considered to be theory of majorizing measures itself, which had the reputation of being opaque and mysterious. As a consequence, most recent treatments of the theory (including by Talagrand himself) have eschewed the use of majorizing measures in favor of a purely combinatorial approach (the generic chaining) where objects based on sequences of partitions provide roughly matching upper and lower bounds on the desired expected supremum. In this paper, we return to majorizing measures as a primary object of study, and give a viewpoint that we think is natural and clarifying from an optimization perspective. As our main contribution, we give an algorithmic proof of the majorizing measures theorem based on two parts: We make the simple (but apparently new) observation that finding the best majorizing measure can be cast as a convex program. This also allows for efficiently computing the measure using off-the-shelf methods from convex optimization. We obtain tree-based upper and lower bound certificates by rounding, in a series of steps, the primal and dual solutions to this convex program. While duality has conceptually been part of the theory since its beginnings, as far as we are aware no explicit link to convex optimization has been previously made.

We consider the task of proving integer infeasibility of a bounded convex K in ℝⁿ using a general branching proof system. In a general branching proof, one constructs a branching tree by adding an integer disjunction 𝐚𝐱 ≤ b or 𝐚𝐱 ≥ b+1, 𝐚 ∈ ℤⁿ, b ∈ ℤ, at each node, such that the leaves of the tree correspond to empty sets (i.e., K together with the inequalities picked up from the root to leaf is empty). Recently, Beame et al (ITCS 2018), asked whether the bit size of the coefficients in a branching proof, which they named stabbing planes (SP) refutations, for the case of polytopes derived from SAT formulas, can be assumed to be polynomial in n. We resolve this question in the affirmative, by showing that any branching proof can be recompiled so that the normals of the disjunctions have coefficients of size at most (n R)^O(n²), where R ∈ ℕ is the radius of an 𝓁₁ ball containing K, while increasing the number of nodes in the branching tree by at most a factor O(n). Our recompilation techniques works by first replacing each disjunction using an iterated Diophantine approximation, introduced by Frank and Tardos (Combinatorica 1986), and proceeds by "fixing up" the leaves of the tree using judiciously added Chvátal-Gomory (CG) cuts. As our second contribution, we show that Tseitin formulas, an important class of infeasible SAT instances, have quasi-polynomial sized cutting plane (CP) refutations. This disproves a conjecture that Tseitin formulas are (exponentially) hard for CP. Our upper bound follows by recompiling the quasi-polynomial sized SP refutations for Tseitin formulas due to Beame et al, which have a special enumerative form, into a CP proof of the same length using a serialization technique of Cook et al (Discrete Appl. Math. 1987). As our final contribution, we give a simple family of polytopes in [0,1]ⁿ requiring exponential sized branching proofs.

In breakthrough work, Tardos (Oper. Res. '86) gave a proximity based framework for solving linear programming (LP) in time depending only on the constraint matrix in the bit complexity model. In Tardos's framework, one reduces solving the LP min(c, x), Ax=b, x ≥ 0, A Zm×n, to solving O(nm) LPs in A having small integer coefficient objectives and right-hand sides using any exact LP algorithm. This gives rise to an LP algorithm in time poly (n, m log ΔA), where ΔAis the largest subdeterminant of A. A significant extension to the real model of computation was given by Vavasis and Ye (Math. Prog. '96), giving a specialized interior point method that runs in time poly (n, m,log χ̑A), depending on Stewart's χ̑A, a well-studied condition number. In this work, we extend Tardos's original framework to obtain such a running time dependence. In particular, we replace the exact LP solves with approximate ones, enabling us to directly leverage the tremendous recent algorithmic progress for approximate linear programming. More precisely, we show that the fundamental “accuracy” needed to exactly solve any LP in A is inverse polynomial in n and log χ̑A. Plugging in the recent algorithm of van den Brand (SODA '20), our method computes an optimal primal and dual solution using O(mnω+1+0(1)log(χ̑A+n)) arithmetic operations, outperforming the specialized interior point method of Vavasis and Ye and its recent improvement by Dadush et al (STOC '20). By applying the preprocessing algorithm of the latter paper, the dependence can also be reduced from χ̑Ato χ̑A*, the minimum value of χ̑ADattainable via column rescalings. Our framework is applicable to achieve the poly (n, m,log χ̑A*) bound using essentially any weakly polynomial LP algorithm, such as the ellipsoid method. At a technical level, our framework combines together approximate LP solutions to compute exact ones, making use of constructive proximity theorems-which bound the distance between solutions of “nearby” LPs-to keep the required accuracy low.

Following the breakthrough work of Tardos (Oper. Res. ’86) in the bit-complexity model, Vavasis and Ye (Math. Prog. ’96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) max c x, Ax = b, x ≥ 0, A ∈ m × n , Vavasis and Ye developed a primal-dual interior point method using a ‘layered least squares’ (LLS) step, and showed that O(n 3.5 log(χ A +n)) iterations suffice to solve (LP) exactly, where χ A is a condition measure controlling the size of solutions to linear systems related to A.

Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng [ J. ACM, 51 (2004), pp. 385--463] who developed the notion of smoothed analysis. Starting from an arbitrary linear program (LP) with $d$ variables and $n$ constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP, known as the smoothed LP, where variance $\sigma^2$ Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected $\widetilde{O}(d^{55} n^{86} \sigma^{-30} + d^{70}n^{86})$ number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by Deshpande and Spielman [ FOCS `05, 2005, pp. 349--356] and later Vershynin [ SIAM J. Comput., 39 (2009), pp. 646--678]. The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected $O\big(\log^2 n \cdot \log\log n \cdot (d^3\sigma^{-4} + d^5\log^2 n + d^9\log^4 d)\big)$ number of pivots, improving the dependence on $n$ from polynomial to polylogarithmic. While the original proof of Spielman and Teng has now been substantially simplified, the resulting analyses are still quite long and complex and the parameter dependencies far from optimal. In this work, we make substantial progress on this front, providing an improved and simpler analysis of shadow simplex methods, where our algorithm requires an expected $O(d^2 \sqrt{\log n} ~ \sigma^{-2} + d^3 \log^{3/2} n)$ number of simplex pivots. We obtain our results via an improved shadow bound, key to earlier analyses as well, combined with improvements on algorithmic techniques of Vershynin. As an added bonus, our analysis is completely modular and applies to a range of perturbations, which, aside from Gaussians, also includes Laplace perturbations.

In this work, we give a novel algorithm for computing dense lattice subspaces, a conjecturally tight characterization of the lattice covering radius, and provide a bound on the slicing constant of lattice Voronoi cells. Our work is motivated by the pursuit of faster algorithms for integer programming, for which we give a conditional speedup based on the recent resolution of the ℓ2 Kannan-Lovász conjecture. Through these results, we hope to motivate further study of the interplay between the recently developed reverse Minkowski theory, lattice algorithms and convex geometry.

We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most $t$ sets. We give an efficient algorithm that finds a coloring with discrepancy $O((t \log n)^{1/2})$, matching the best known nonconstructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an $O(t^{1/2} \log n)$ bound. The result also extends to the more general Komlós setting and gives an algorithmic $O(\log^{1/2} n)$ bound.

In this paper, we show that any sequence of infinite lattice constellations which is good for the unconstrained Gaussian channel can be shaped into a capacity-achieving sequence of codes for the power-constrained Gaussian channel under lattice decoding and non-uniform signaling. Unlike previous results in the literature, our scheme holds with no extra condition on the lattices (e.g., quantization-goodness or vanishing flatness factor), thus establishing a direct implication between AWGN-goodness, in the sense of Poltyrev and capacity-achieving codes. Our analysis uses properties of the discrete Gaussian distribution in order to obtain precise bounds on the probability of error and achievable rates. In particular, we obtain a simple characterization of the finite-blocklength behavior of the scheme, showing that it approaches the optimal dispersion coefficient for high signal-to-noise ratio. We further show that for low signal-to-noise ratio, the discrete Gaussian over centered lattice constellations cannot achieve capacity, and thus a shift (or “dither”) is essentially necessary.

In the vector balancing problem, we are given symmetric convex bodies C and K in R^n, and our goal is to determine the minimum number β ≥ 0, known as the vector balancing constant from C to K, such that for any sequence of vectors in C there always exists a signed combination of them lying inside β K. Many fundamental results in discrepancy theory, such as the Beck-Fiala theorem (Discrete Appl.~Math '81), Spencer's "six standard deviations suffice" theorem (Trans.~Amer.~Math.~Soc '85) and Banaszczyk's vector balancing theorem (Random Structures & Algorithms '98) correspond to bounds on vector balancing constants. The above theorems have inspired much research in recent years within theoretical computer science. In this work, we show that all vector balancing constants admit "good" approximate characterizations, with approximation factors depending only polylogarithmically on the dimension n. First, we show that a volumetric lower bound due to Banaszczyk is tight within a O(log n) factor. Our proof is algorithmic, and we show that Rothvoss's (FOCS '14) partial coloring algorithm can be analyzed to obtain these guarantees. Second, we present a novel convex program which encodes the "best possible way" to apply Banaszczyk's vector balancing theorem for bounding vector balancing constants from above, and show that it is tight within an O(log^2.5 n) factor. This also directly yields a corresponding polynomial time approximation algorithm both for vector balancing constants, and for the hereditary discrepancy of any sequence of vectors with respect to an arbitrary norm.

Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC'98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes "nearby" points to the same bucket and "far away" points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps determine both query time and space usage. In a seminal work, Andoni and Indyk (FOCS '06) constructed an LSH family based on random ball partitionings of space that achieves an LSH exponent of 1/c^2 for the l_2 norm, which was later shown to be optimal by Motwani, Naor and Panigrahy (SIDMA '07) and O'Donnell, Wu and Zhou (TOCT '14). Although optimal in the LSH exponent, the ball partitioning approach is computationally expensive. So, in the same work, Andoni and Indyk proposed a simpler and more practical hashing scheme based on Euclidean lattices and provided computational results using the 24-dimensional Leech lattice. However, no theoretical analysis of the scheme was given, thus leaving open the question of finding the exponent of lattice based LSH. In this work, we resolve this question by showing the existence of lattices achieving the optimal LSH exponent of 1/c^2 using techniques from the geometry of numbers. At a more conceptual level, our results show that optimal LSH space partitions can have periodic structure. Understanding the extent to which additional structure can be imposed on these partitions, e.g. to yield low space and query complexity, remains an important open problem.

We provide a deterministic construction of the sparse Johnson-Lindenstrauss transform of Kane & Nelson (J.ACM 2014) which runs, under a mild restriction, in the time necessary to apply the sparse embedding matrix to the input vectors.Specifically, given a set of n vectors in R d and target error ε, we give a deterministic algorithm to compute a {-1, 0, 1} embedding matrix of rank O((ln n)/ε 2 ) with O((ln n)/ε) entries per column which preserves the norms of the vectors to within 1±ε.If NNZ, the number of non-zero entries in the input set of vectors, is Ω(d 2 ), our algorithm runs in time O(NNZ • ln n/ε).One ingredient in our construction is an extremely simple proof of the Hanson-Wright inequality for subgaussian random variables, which is more amenable to derandomization.As an interesting byproduct, we are able to derive the essentially optimal form of the inequality in terms of its functional dependence on the parameters.

We present a new class of polynomial-time algorithms for submodular function minimization (SFM), as well as a unified framework to obtain strongly polynomial SFM algorithms. Our new algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and the Fujishige-Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. Firstly, we use the geometric rescaling technique, which has recently gained attention in linear programming. We adapt this technique to SFM and obtain a weakly polynomial bound O((n4 · EO + n5) log(nL)). Secondly, we exhibit a general combinatorial black-box approach to turn any strongly polynomial εL-approximate SFM oracle into an strongly polynomial exact SFM algorithm. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudopolynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige-Wolfe algorithm. Combined with the geometric rescaling technique, the black-box approach provides a O((n5 · EO + n6) log2 n) algorithm. Finally, we show that one of the techniques we develop in the paper, “sliding”, can also be combined with the cutting-plane method of Lee, Sidford, and Wong [27], yielding a simplified variant of their O(n3 log2 n · EO + n4 logO(1) n) algorithm.

An important result in discrepancy due to Banaszczyk states that for any set of n vectors in ℝm of ℓ2 norm at most 1 and any convex body K in ℝm of Gaussian measure at least half, there exists a ± 1 combination of these vectors which lies in 5K. This result implies the best known bounds for several problems in discrepancy. Banaszczyk’s proof of this result is non-constructive and an open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors.

Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng (JACM ‘04), who the developed the notion of smoothed analysis. Starting from an arbitrary linear program with d variables and n constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP (smoothed LP), where variance σ Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected O(n86 d55 σ−30) number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by SpielmanDeshpande (FOCS ‘05) and later Vershynin (SICOMP ‘09). The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected O(d3 log3 n σ−4 + d9log7 n) number of pivots, improving the dependence on n from polynomial to logarithmic.

An important theorem of Banaszczyk (Random Structures & Algorithms 1998) states that for any sequence of vectors of l_2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R^n, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination. We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric. As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/sqrt{log n}), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)-subgaussian signing distributions when the vectors have length O(1/sqrt{log n}), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies.

We introduce and study the Lattice Distortion Problem (LDP). LDP asks how "similar" two lattices are. I.e., what is the minimal distortion of a linear bijection between the two lattices? LDP generalizes the Lattice Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply asks whether the minimal distortion is one. As our first contribution, we show that the distortion between any two lattices is approximated up to a n^{O(log(n))} factor by a simple function of their successive minima. Our methods are constructive, allowing us to compute low-distortion mappings that are within a 2^{O(n*log(log(n))/log(n))} factor of optimal in polynomial time and within a n^{O(log(n))} factor of optimal in singly exponential time. Our algorithms rely on a notion of basis reduction introduced by Seysen (Combinatorica 1993), which we show is intimately related to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to within any constant factor (under randomized reductions), by a reduction from the Shortest Vector Problem.

We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an efficient algorithm that finds a coloring with discrepancy O((t log n)1/2), matching the best known non-constructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t1/2log n) bound. Our result also extends to the more general Komlós setting and gives an algorithmic O(log1/2n) bound.

We present a natural reverse Minkowski-type inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits a surprising wealth of connections to various areas in mathematics and computer science, including a conjecture motivated by integer programming by Kannan and Lovasz (Annals of Math. 1988), a question from additive combinatorics asked by Green, a question on Brownian motions asked by Saloff-Coste (Colloq. Math. 2010), a theorem by Milman and Pisier from convex geometry (Ann. Probab. 1987), worst-case to average-case reductions in lattice-based cryptography, and more. We present these connections, provide evidence for the conjecture, and discuss possible approaches towards a proof. Our main technical contribution is in proving that our conjecture implies the l2 case of the Kannan and Lovasz conjecture. The proof relies on a novel convex relaxation for the covering radius, and a rounding procedure based on "uncrossing" lattice subspaces.

We give a 2^{O(n)}(1+1/eps)^n time and poly(n)-space deterministic algorithm for computing a (1+eps)^n approximation to the volume of a general convex body K, which comes close to matching the (1+c/eps)^{n/2} lower bound for volume estimation in the oracle model by Barany and Furedi (STOC 1986, Proc. Amer. Math. Soc. 1988). This improves on the previous results of Dadush and Vempala (Proc. Nat'l Acad. Sci. 2013), which gave the above result only for symmetric bodies and achieved a dependence of 2^{O(n)}(1+log^{5/2}(1/eps)/eps^3)^n. For our methods, we reduce the problem of volume estimation in K to counting lattice points in K subseteq R^n (via enumeration) for a specially constructed lattice L: a so-called thin covering of space with respect to K (more precisely, for which L + K = R^n and vol_n(K)/det(L) = 2^{O(n)}). The trade off between time and approximation ratio is achieved by scaling down the lattice. As our main technical contribution, we give the first deterministic 2^{O(n)}-time and poly(n)-space construction of thin covering lattices for general convex bodies. This improves on a recent construction of Alon et al (STOC 2013) which requires exponential space and only works for symmetric bodies. For our construction, we combine the use of the M-ellipsoid from convex geometry (Milman, C.R. Math. Acad. Sci. Paris 1986) together with lattice sparsification and densification techniques (Dadush and Kun, SODA 2013; Rogers, J. London Math. Soc. 1950).

We study the simplex method over polyhedra satisfying certain "discrete curvature" lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al. (SOCG 2012), Brunsch and Röglin (ICALP 2013), and Eisenbrand and Vempala (2014) have improved our understanding of such polyhedra. We develop a new type of dual analysis of the shadow simplex method which provides a clean and powerful tool for improving all previously mentioned results. Our methods are inspired by the recent work of Bonifas and the first named author, who analyzed a remarkably similar process as part of an algorithm for the Closest Vector Problem with Preprocessing. For our first result, we obtain a constructive diameter bound of O((n^2 / delta) ln (n / delta)) for n-dimensional polyhedra with curvature parameter delta in (0, 1]. For the class of polyhedra arising from totally unimodular constraint matrices, this implies a bound of O(n^3 ln n). For linear optimization, given an initial feasible vertex, we show that an optimal vertex can be found using an expected O((n^3 / delta) ln (n / delta)) simplex pivots, each requiring O(mn) time to compute. An initial feasible solution can be found using O((mn^3 / delta) ln (n / delta)) pivot steps.

We give a 2n+o(n)-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm, the deterministic Õ(4n)-time and Õ(2n)-space algorithm of Micciancio and Voulgaris [1]. We achieve our main result in three steps. First, we show how to modify the sampling algorithm from [2] to solve the problem of discrete Gaussian sampling over lattice shifts, L - t, with very low parameters. While the actual algorithm is a natural generalization of [2], the analysis uses substantial new ideas. This yields a 2n+o(n)-time algorithm for approximate CVP with the very good approximation factor γ = 1 + 2-o(n/ log n). Second, we show that the approximate closest vectors to a target vector t can be grouped into “lower-dimensional clusters,” and we use this to obtain a recursive reduction from exact CVP to a variant of approximate CVP that “behaves well with these clusters.” Third, we show that our discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP. The analysis depends crucially on some new properties of the discrete Gaussian distribution and approximate closest vectors, which might be of independent interest.

Improving on the Voronoi cell based techniques of [28, 24], we give a Las Vegas Õ (2n) expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the Õ(4n) deterministic runtime of the Micciancio Voulgaris algorithm [24] (henceforth MV) for CVPP1 at the cost of a polynomial amount of randomness (which only affects runtime, not correctness). As in MV, our algorithm proceeds by computing a short path on the Voronoi graph of the lattice, where lattice points are adjacent if their Voronoi cells share a common facet, from the origin to a closest lattice vector. Our main technical contribution is a randomized procedure that, given the Voronoi relevant vectors of a lattice – the lattice vectors inducing facets of the Voronoi cell – as preprocessing, and any “close enough” lattice point to the target, computes a path to a closest lattice vector of expected polynomial size. This improves on the Õ(2n) path length given by the MV algorithm. Furthermore, as in MV, each edge of the path can be computed using a single iteration over the Voronoi relevant vectors. As a byproduct of our work, we also give an optimal relationship between geometric and path distance on the Voronoi graph, which we believe to be of independent interest.

We give a randomized 2n+o(n)-time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic ~O(4n)-time and ~O(2n)-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample 2n/2 vectors from the discrete Gaussian distribution at any parameter in 2n+o(n) time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS.

We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n/√log n), improving on the previous best of O(n1.5) due to Lag arias, Lenstra, and Schnorr [1]. We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(√n/log n) due to Aharonov and Regev [2]. To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio [3] gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+∈)γ approximate CVP to γ approximate CVP where the target is at distance at most 1+1/∈ times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun [4]. As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice [5]. We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide Ω(√n) factor larger.

The smoothing parameter ηε(L) of a Euclidean lattice L, introduced by Micciancio and Regev (FOCS'04; SICOMP'07), is (informally) the smallest amount of Gaussian noise that “smooths out” the discrete structure of L (up to error ε). It plays a central role in the best known worst-case/average-case reductions for lattice problems, a wealth of lattice-based cryptographic constructions, and (implicitly) the tightest known transference theorems for fundamental lattice quantities. In this work we initiate a study of the complexity of approximating the smoothing parameter to within a factor γ, denoted γ-GapSPP. We show that (for ε = 1/ poly(n)): . (2+o(1))-GapSPP ∈ AM, via a Gaussian analogue of the classic Goldreich-Goldwasser protocol (STOC'98); . (1 + o(1))-GapSPP ∈ coAM, via a careful application of the Goldwasser-Sipser (STOC'86) set size lower bound protocol to thin shells in Rn; . (2 + o(1))-GapSPP E SZK ⊆ AM ∩ coAM (where SZK is the class of problems having statistical zero-knowledge proofs), by constructing a suitable instance-dependent commitment scheme (for a slightly worse o(1)-term); . (1 + o(1))-GapSPP can be solved in deterministic 2O(n)polylog(1/ε) time and 2O(n)space. As an application, we demonstrate a tighter worst-case to average-case reduction for basing cryptography on the worstcase hardness of the GapSPP problem, with Õ(√n) smaller approximation factor than the GapSVP problem. Central to our results are two novel, and nearly tight, characterizations of the magnitude of discrete Gaussian sums over L: the first relates these directly to the Gaussian measure of the Voronoi cell of L, and the second to the fraction of overlap between Euclidean balls centered around points of L.

We give a deterministic algorithm for solving the (1 + ε) approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2O(n)(1 + 1/ε)n time and 2n poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the “AKS Sieve” based algorithms for (1 + ε)-CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the existence of a poly(n)-space and 2O(n) time algorithm for exact CVP in the l2 norm, the space complexity of our algorithm can be reduced to polynomial. Our main technical contribution is a method for “sparsifying” any input lattice while approximately maintaining its metric structure. To this end, we employ the idea of random sublattice restrictions, which was first employed by Khot (FOCS 2003) for the purpose of proving hardness for Shortest Vector Problem (SVP) under lp norms.

We give algorithms for computing the densest k-dimensional sublattice of an arbitrary lattice, and related problems. This is an important problem in the algorithmic geometry of numbers that includes as special cases Rankin's problem (which corresponds to the densest sublattice problem with respect to the Euclidean norm, and has applications to the design of lattice reduction algorithms), and the shortest vector problem for arbitrary norms (which corresponds to setting k = 1) and its dual (k = n − 1). Our algorithm works for any norm and has running time kO(k · n) and uses 2n poly(n) space. In particular, the algorithm runs in single exponential time 2O(n) for any constant k = O(1).

We give a deterministic O(log n)n-time and space algorithm for the Shortest Vector Problem (SVP) of a lattice under any norm, improving on the previous best deterministic nO(n)-time algorithms for general norms. This approaches the 2O(n)-time and space complexity of the randomized sieve based SVP algorithms (Arvind and Joglekar, FSTTCS 2008), first introduced by Ajtai, Kumar and Sivakumar (STOC 2001) for ℓ2-SVP, and the M-ellipsoid covering based SVP algorithm of Dadush et al. (FOCS 2011).

We investigate the problem of designing differentially private mechanisms for a set of d linear queries over a database, while adding as little error as possible. Hardt and Talwar [HT10] related this problem to geometric properties of a convex body defined by the set of queries and gave a O(log3 d)-approximation to the minimum l22 error, assuming a conjecture from convex geometry called the Slicing or Hyperplane conjecture. In this work we give a mechanism that works unconditionally, and also gives an improved O(log2 d) approximation to the expected l22 error. We remove the dependence on the Slicing conjecture by using a result of Klartag [Kla06] that shows that any convex body is close to one for which the conjecture holds; our main contribution is in making this result constructive by using recent techniques of Dadush, Peikert and Vempala [DPV10]. The improvement in approximation ratio relies on a stronger lower bound we derive on the optimum. This new lower bound goes beyond the packing argument that has traditionally been used in Differential Privacy and allows us to add the packing lower bounds obtained from orthogonal subspaces. We are able to achieve this via a symmetrization argument which argues that there always exists a near optimal differentially private mechanism which adds noise that is independent of the input database! We believe this result should be of independent interest, and also discuss some interesting consequences.

We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming (IP). Our enumeration technique relies on a classical concept from asymptotic convex geometry known as the M-ellipsoid, and uses as a crucial subroutine the recent algorithm of Micciancio and Voulgaris (STOC 2010)for lattice problems in the ℓ2norm. As a main technical contribution, which may be of independent interest, we build on the techniques of Klartag (Geometric and Functional Analysis, 2006) to give an expected 2O(n)-time algorithm for computing an M-ellipsoid for any n-dimensional convex body. As applications, we give deterministic 2O(n)-time and -space algorithms for solving exact SVP, and exact CVP when the target point is sufficiently close to the lattice, on n-dimensional lattices in any (semi-)norm given an M-ellipsoid of the unit ball. In many norms of interest, including all ℓpnorms, an M-ellipsoid is computable in deterministic poly(n) time, in which case these algorithms are fully deterministic. Here our approach may be seen as a derandomization of the "AKS sieve" for exact SVP and CVP (Ajtai, Kumar, and Siva Kumar, STOC2001 and CCC 2002). As a further application of our SVP algorithm, we derive an expected O(f*(n))n-time algorithm for Integer Programming, where f*(n) denotes the optimal bound in the so-called "flatnesstheorem," which satisfies f*(n) = O(n4/3polylog(n))and is conjectured to be f*(n) = O(n). Our runtime improves upon the previous best of O(n2)nby Hildebrand and Koppe (2010).

Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows polynomially in the dimension and inverse polynomially in the fraction of the volume taken up by the kernel of the star-shaped body. The analysis is based on a new isoperimetric inequality. Our main technical contribution is a tool for proving such inequalities when the domain is not convex. As a consequence, we obtain a polynomial algorithm for computing the volume of such a set as well. In contrast, linear optimization over star-shaped sets is NP-hard.