We study a new approach for constructing pseudorandom generators (PRGs) that fool constant-width standard-order read-once branching programs (ROBPs). Let X be the n-bit output distribution of the INW PRG (Impagliazzo, Nisan, and Wigderson, STOC 1994), instantiated using expansion parameter.. We prove that the bitwise XOR of t independent copies of X fools width- w programs with error n(log(w+1)) center dot (lambda center dot log n)(t). Notably, this error bound is meaningful even for relatively large values of lambda such as lambda = 1/O(log n). Admittedly, our analysis does not yet imply any improvement in the bottom-line overall seed length required for fooling such programs - it just gives a new way of re-proving the well-known O(log(2) n) bound. Furthermore, we prove that this shortcoming is not an artifact of our analysis, but rather is an intrinsic limitation of our "XOR of INW" approach. That is, no matter how many copies of the INW generator we XOR together, and no matter how we set the expansion parameters, if the generator fools width-3 programs and the proof of correctness does not use any properties of the expander graphs except their spectral expansion, then we prove that the seed length of the generator is inevitably Omega(log(2) n). Still, we hope that our work might be a step toward constructing near-optimal PRGs fooling constant-width ROBPs. We suggest that one could try running the INW PRG on t correlated seeds, sampled via another PRG, and taking the bitwise XOR of the outputs.
We study the problem of transforming an algorithm for matrix multiplication, whose output has a small fraction of the entries correct into a matrix multiplication algorithm, whose output is fully correct for all inputs. In this work, we provide a new and simple way to transform an average-case algorithm that takes two matrices A, B epsilon F-nxn (p) for a prime p, and outputs a matrix that agrees with the matrix product AB on a 1/ p + epsilon fraction of entries on average for a small epsilon > 0, into a worst-case algorithm that correctly computes the matrix product for all possible inputs. Our reduction employs list-decodable codes to transform an average-case algorithm into an algorithm with one-sided error, which are known to admit efficient reductions from the work of Gola, Shinkar, and Singh [12]. Our reduction is more concise and straightforward compared to the recent work of Hirahara and Shimizu [18], and improves the overhead in the running time incurred during the reduction.
We explore fundamental geometric computations on point sets that are given to the algorithm implicitly. In particular, we are given a database which is a collection of tables with numerical values, and the geometric computation is to be performed on the join of the tables. Explicitly computing this join takes time exponential in the size of the tables. We are therefore interested in geometric problems that can be solved by algorithms whose running time is a polynomial in the size of the tables. Such relational algorithms are typically not able to explicitly compute the join. To avoid the NP-completeness bottleneck, researchers assume that the tables have a tractable combinatorial structure, like being acyclic. Even with this assumption, simple geometric computations turn out to be non-trivial and sometimes intractable. In this article, we study the problem of computing the maximum distance of a point in the join to a given subspace, and develop approximation algorithms for this NP-hard problem.
Angluin (STOC'80) and Yamashita and Kameda (PODC'88) show that some useful distributed tasks are impossible (for deterministic algorithms) in a general network if nodes do not possess unique identifiers. However, any task decidable in the non-distributed context, can be solved deterministically if the network has a unique leader. Alternatively, much research has been devoted to randomized distributed algorithms in anonymous networks. We present tight upper and lower bounds for the fundamental question: How much randomness is necessary and sufficient to solve Leader Election (LE) in anonymous networks, i.e., to transform an anonymous network into a non-anonymous one? We prove that at least one random bit per node is required in some cases. Surprisingly, a single random bit is also enough, for a total of n bits, where n is the number of nodes. However, the time complexity of our (total of) n random bits algorithm for general networks turned out to be impractically high. Hence, we also developed time-efficient algorithms for the very symmetric graphs of cliques and cycles, paying only an additional cost of o(n) random bits. The primary steps of our algorithms are of independent interest. At first glance, it seems that using one random bit per node, any algorithm can distinguish only two sets of nodes: those with 0 and those with 1. Our algorithms manage to partition the nodes into more than two sets with high probability. In some sense, they perform the task of a "distributed pseudorandom generator", for example, one of our algorithms turns n bits, one per node, into n unique (with high probability) numbers. Even though a complete graph looks very symmetric, the algorithms explore interesting asymmetries inherent in any n permutations (of n values each), if each describes the assignment (by the adversary) of ports in a node to edges leading to neighbors. Finally, we show how to transform any randomized algorithm that generates xn + o(n) random bits in total to one where each node generates at most x + 1 bits. Our results apply to both synchronous and asynchronous networks.
The advertisement placement problem involves selecting and scheduling ads within a timeline that has capacity constraints to maximize profit. Each task is characterized by its height, width, and profit, and must be fully scheduled across multiple time slots. This problem models practical scenarios such as internet advertising and energy management, and it also generalizes classical combinatorial optimization problems like the knapsack and bin packing problems. We present a simple (2 + epsilon)-approximation algorithm for any epsilon > 0, which improves upon the state-of-the-art 3 + epsilon factor established by Freund and Naor twenty years ago. Our approach combines rounding techniques with dynamic programming and an efficient extension of list scheduling. Furthermore, we enhance this method with linear programming techniques to provide an almost optimal (1 + epsilon)-approximation algorithm under resource augmentation, which allows for a slight increase in time slot capacities.
Given a circuit C : {0, 1}(n) -> {0, 1}(m) from a circuit class C, with m > n, finding a y is an element of {0, 1}(m) such that for all x {0, 1}(n), C(x) not equal y, is the range avoidance problem (denoted by C-AVOID). Deterministic polynomial time algorithms (even with access to NP oracles) solving this problem are known to imply explicit constructions of various pseudorandom objects like hard Boolean functions, linear codes, PRGs etc. Deterministic polynomial time algorithms are known for NC20-Avoid when m > n, and for NC30-Avoid when m >= n(2)/log n, where NCk0 is the class of circuits with bounded fan-in which have constant depth and the output depends on at most k of the input bits. On the other hand, it is also known that NC30-Avoid when m = n + O (n2/3) is at least as hard as explicit construction of rigid matrices. In fact, algorithms for solving range avoidance for even NC40 circuits imply new circuit lower bounds. In this paper, we propose a new approach to solving the range avoidance problem via hypergraphs. We formulate the problem in terms of Turan-type problems in hypergraphs of the following kind: for a fixed k-uniform hypergraph H, what is the maximum number of edges that can exist in HC, which does not have a sub-hypergraph isomorphic to H? We show the following: We first demonstrate the applicability of this approach by showing alternate proofs of some of the known results for the range avoidance problem using this framework. We then use our approach to show (using several different hypergraph structures for which Turan-type bounds are known in the literature) that there is a constant c such that MONOTONE-NC30-Avoid can be solved in deterministic polynomial time when m > cn(2). To improve the stretch constraint to linear, more precisely, to m > n, we show a new Turan-type theorem for a hypergraph structure (which we call the loose X-2l-cycles). More specifically, we prove that any connected 3-uniform linear hypergraph with m > n edges must contain a loose X-2l cycle. This may be of independent interest. Using this, we show that MONOTONE-NC30-Avoid can be solved in deterministic polynomial time when m > n, thus improving the known bounds of NC30-Avoid for the case of monotone circuits. In contrast, we note that efficient algorithms for solving MONOTONE-NC60-Avoid, already imply explicit constructions for rigid matrices. We also generalise our argument to solve the special case of range avoidance for NCk0 where each output function computed by the circuit is the majority function on its inputs, where m > n(2).
We propose a simple variant of the INW pseudo-random generator, where blocks have varying lengths, and prove it gives the same parameters as the more complicated construction of Armoni's PRG. This shows there is no need for the specialized PRGs of Nisan and Zuckerman and Armoni, and they can be obtained as simple variants of INW. For the construction to work we need space-efficient extractors with tiny entropy loss. We use the extractors from [2] instead of [6] taking advantage of the very high min-entropy regime we work with. We remark that using these extractors has the additional benefit of making the dependence on the branching program alphabet Sigma correct.
In this paper, we show that random Gabidulin codes of block length n and rate R achieve the (average-radius) list decoding capacity of radius 1 - R - epsilon in the rank metric with an order-optimal column-to-row ratio of O(epsilon). This extends the recent work of Guo, Xing, Yuan, and Zhang (FOCS 2024), improving their column-to-row ratio from O( (epsilon) (n)) to O(epsilon). For completeness, we also establish a matching lower bound on the column-to-row ratio for capacity-achieving Gabidulin codes in the rank metric. Our proof techniques build on the work of Guo and Zhang (FOCS 2023), who showed that randomly punctured Reed-Solomon codes over fields of quadratic size attain the generalized Singleton bound of Shangguan and Tamo (STOC 2020) in the Hamming metric. The proof of our lower bound follows the method of Alrabiah, Guruswami, and Li (SODA 2024) for codes in the Hamming metric.
This paper is motivated by basic complexity and probability questions about permanents of random matrices over small finite fields, and in particular, about properties separating the permanent and the determinant. Let q be a fixed odd prime, and let k <= n both be growing. For a uniformly random nxk matrix A over F-q, we study the probability that all k x k submatrices of A have zero permanent; namely that A does not have full permanental rank. When k = n, this is simply the probability that a random square matrix over F-q has zero permanent, which we do not understand. We believe that the probability in this case is (1) (q) + o(1), which would be in contrast to the case of the determinant, where the answer is (1) (q) + Omega (q) (1). Our main result is that when k is O(root n), the probability that a random n x k matrix does not have full permanental rank is essentially the same as the probability that the matrix has a 0 column, namely (1 + o(1)) (k) q(n). In contrast, for determinantal (standard) rank the analogous probability is Theta( (qk) (qn)). At the core of our result are some basic linear algebraic properties of the permanent that distinguish it from the determinant.
In this work, we study k-min-sum-of-radii (k-MSR) clustering under mergeable constraints. k-MSR seeks to group data points using a set of up to k balls, such that the sum of the radii of the balls is minimized. A clustering constraint is called mergeable if merging two clusters satisfying the constraint, results in a cluster that also satisfies the constraint. Many popularly studied constraints are mergeable, including fairness constraints and lower bound constraints. In our work, we design a (4 +epsilon)-approximation for k-MSR under any given mergeable constraint with runtime 2O( k/epsilon center dot log(2) k/epsilon)n(4), i.e., fixed-parameter tractable in k for constant epsilon. Our result directly improves upon the FPT (6 +epsilon)-approximation by Carta et al. [10]. We also provide a hardness result that excludes the exact solvability of k-MSR under any given mergeable constraint in time f(k)n(o(k)), assuming ETH is true.
In this work we study online optimization problems with congestion effects. These are problems where tasks arrive online and a decision maker is required to allocate them on the fly to available resources in order to minimize the cost suffered, which grows with the amount of resources used. This class of problems corresponds to the online counterpart of well-known studied problems, including optimization problems with diseconomies of scale [33], minimum cost in congestion games [25], and load balancing problems [4]. Within this setting, our work settles the problem of designing online algorithms with optimal competitive ratio, i.e., algorithms whose incurred cost is as close as possible to that of an oracle with complete knowledge of the future instance ahead of time. We provide three contributions underpinning this result. First, we show that no online algorithm can achieve a competitive ratio below a given factor depending solely on the resource costs. Second, we show that, when guided by carefully modified cost functions, the greedy algorithm achieves a competitive ratio matching this lower bound and thus is optimal. Finally, we show how to compute such modified cost functions in polynomial time.
Maxcut is a fundamental problem in graph algorithms, extensively studied for its theoretical and practical significance. The goal is to partition the vertex set of a graph G = ( V, E) into disjoint subsets S and V \ S so as to maximize the number of edges crossing the cut ( S, V \ S). The seminal work of Goemans and Williamson [39] introduced a semidefinite programming (SDP) based algorithm achieving a alpha(GW) similar to 0.87856-approximation for general graphs, guaranteed to be optimal under the Unique Games Conjecture [56, 57]. We revisit the Goemans-Williamson SDP and prove that the standard Maxcut SDP achieves a ( alpha(GW) + Omega(1))-approximation whenever the input graph contains Omega(|E|) edge-disjoint triangles. Our analysis builds on classical rounding techniques studied in [39, 76] and introduces a refined understanding of the SDP solution structure in regimes where the previous guarantees are tight. Our result identifies a simple combinatorial property that may be satisfied by many natural graph classes. As applications, we show that unit ball graphs and graphs satisfying a spectral transitivity condition (as studied in [40, 12]) meet our structural criterion, and therefore we get better than aGW approximation guarantees for them. Our algorithm runs in nearly linear time O(|E|), offering a more practical alternative to the PTAS of [48] for unit ball graphs, which has exponential dependence on the approximation parameter.
We design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong lower bound results obtained over this algebra by Hrubes, Yehudayoff, and Wigderson [12] and Fijalkow, Lagarde, Ohlmann, and Serre [9] from the identity testing perspective. Our main results are the following: We construct nonassociative algebras (both commutative and noncommutative) which have no low degree identities. As a result, we obtain the first Amitsur-Levitzki type theorems [2] over nonassociative polynomial algebras. As a direct consequence, we obtain randomized polynomialtime black-box PIT algorithms for nonassociative polynomials which allow evaluation over such algebras. On the derandomization side, we give a deterministic polynomial-time identity testing algorithm for nonassociative polynomials given by arithmetic circuits in the white-box setting. Previously, such an algorithm was known with the additional restriction of noncommutativity [4]. In the black-box setting, we construct a hitting set of quasipolynomial-size for nonassociative polynomials computed by arithmetic circuits of small depth. Understanding the black-box complexity of identity testing, even in the randomized setting, was open prior to our work.
We study the problem of strongly refuting semirandom k-LIN(F) instances: systems of k-sparse inhomogeneous linear equations over a finite field F. For the case of F = F-2, this is the well-studied problem of refuting semirandom instances of k-XOR, where the works of [18, 20] establish a tight trade-off between runtime and clause density for refutation: for any choice of a parameter l, they give an n(O(l))-time algorithm to certify that there is no assignment that can satisfy more than 1/2 + epsilon-fraction of constraints in a semirandom k-XOR instance, provided that the instance has O(n) center dot (n/l)(k/2-1) log n/epsilon(4) constraints, and the work of [28] provides good evidence that this tight up to a polylog(n) factor via lower bounds for the Sum-of-Squares hierarchy. However, for larger fields, the only known results for this problem are established via black-box reductions to the case of F-2, resulting in a |F|(3k) gap between the current best upper and lower bounds. In this paper, we give an algorithm for refuting semirandom k-LIN(F) instances with the "correct" dependence on the field size |F|. For any choice of a parameter l, our algorithm runs in (|F|n)(O(l))-time and strongly refutes semirandom k-LIN(F) instances with at least O(n) center dot (|F*|n/l)(k/2-1) log( n|F*|)/epsilon(4) constraints. We give good evidence that this dependence on the field size |F| is optimal by proving a lower bound for the Sum-of-Squares hierarchy that matches this threshold up to a polylog( n|F*|) factor. Our results also extend beyond finite fields to the more general case of Zm and arbitrary finite Abelian groups. Our key technical innovation is a generalization of the "F-2 Kikuchi matrices" of [36, 18] to larger fields, and finite Abelian groups more generally.
We give polynomial time logarithmic approximation guarantees for the budget minimization, as well as for the profit maximization versions of minimum spanning tree interdiction. In this problem, the goal is to remove some edges of an undirected graph with edge weights and edge costs, so as to increase the weight of a minimum spanning tree. In the budget minimization version, the goal is to minimize the total cost of the removed edges, while achieving a desired increase. in the weight of the minimum spanning tree. An alternative objective within the same framework is to maximize the profit of interdiction, namely the increase in the weight of the minimum spanning tree, subject to a budget constraint. There are known polynomial time O(1) approximation guarantees for a similar objective (maximizing the total cost of the tree, rather than the increase). However, the guarantee does not seem to apply to the increase in cost. Moreover, the same techniques do not seem to apply to the budget version. Our approximation guarantees are motivated by studying the question of minimizing the cost of increasing the minimum spanning tree by any amount. We show that in contrast to the budget and profit problems, this version of interdiction is polynomial time-solvable, and we give an efficient algorithm for solving it. The solution motivates a graph-theoretic relaxation of the NP-hard interdiction problem. The gain in minimum spanning tree weight, as a function of the set of removed edges, is super-modular. Thus, the budget problem is an instance of minimizing a linear function subject to a super-modular covering constraint. We use the graph-theoretic relaxation to design and analyze a batch greedy-based algorithm.
In this extended abstract, we show that the max entropy algorithm is a randomized 1.49776 approximation for half-integral TSP, improving upon the previous known bound of 1.49993 from Karlin et al. This also improves upon the best-known approximation for half-integral TSP due to Gupta et al. Our improvement results from using the dual, instead of the primal, to analyze the expected cost of the matching. We believe this method of analysis could lead to a simpler proof that max entropy is a better-than-3/2 approximation in the general case.
A long line of work has studied the pseudorandomness properties of walks on expander graphs. A central goal is to measure how closely the distribution over n-length walks on an expander approximates the uniform distribution of n-independent elements. One approach to do so is to label the vertices of an expander with elements from an alphabet Sigma, and study closeness of the mean of functions over Sigma(n), under these two distributions. We say expander walks epsilon-fool a function if the expander walk mean is epsilon-close to the true mean. There has been a sequence of works studying this question for various functions, such as the XOR function, the AND function, etc. We show that: The class of symmetric functions is O(|Sigma|lambda)-fooled by expander walks over any generic.-expander, and any alphabet Sigma. This generalizes the result of Cohen, Peri, Ta-Shma [STOC'21] which analyzes it for |Sigma| = 2, and exponentially improves the previous bound of O(|Sigma|(O(|Sigma|))lambda), by Golowich and Vadhan [CCC'22]. Moreover, if the expander is a Cayley graph over Z(|Sigma|), we get a further improved bound of O(root|Sigma|lambda). Morever, when Sigma is a finite group G, we show the following for functions over Gn: The class of symmetric class functions is O (root|G|/D lambda)-fooled by expander walks over "structured" lambda-expanders, if G is D-quasirandom. We show a lower bound of Omega(lambda) for symmetric functions for any finite group G (even for "structured" lambda-expanders). We study the Fourier spectrum of a class of non-symmetric functions arising from word maps, and show that they are exponentially fooled by expander walks. Our proof employs Fourier analysis over general groups, which contrasts with earlier works that have studied either the case of Z(2) or Z. This enables us to get quantitatively better bounds even for unstructured sets.
We study the problem of solving linear programs of the form Ax <= b, x <= 0 with differential privacy. For homogeneous LPs Ax >= 0, we give an efficient (epsilon, delta)-differentially private algorithm which with probability at least 1 - beta finds in polynomial time a solution that satisfies all but O( (d2)(epsilon) log(2) (d) (delta beta) root log (1)(rho 0)) constraints, for problems with margin (rho 0) > 0. This improves the bound of O( d(5) (epsilon) log(1.5) (1)(rho 0) poly log( d, (1) (delta), (1) (beta))) by [Kaplan-Mansour-Moran-Stemmer-Tur, STOC '25]. For general LPs Ax <= b, x >= 0 with potentially zero margin, we give an efficient (epsilon, delta)-differentially private algorithm that w.h.p drops O( (d4)(epsilon) log(2.5) (d) (delta) root log dU) constraints, where U is an upper bound for the entries of A and b in absolute value. This improves the result by Kaplan et al. by at least a factor of d(5). Our techniques build upon privatizing a rescaling perceptron algorithm by [Hoberg-Rothvoss, IPCO '17] and a more refined iterative procedure for identifying equality constraints by Kaplan et al.
We study the classic Max-Cut problem under multiple cardinality constraints, which we refer to as the Constrained Max-Cut problem. Given a graph G = ( V, E), a partition of the vertices into c disjoint parts V-1,..., V-c, and cardinality parameters k(1),..., k(c), the goal is to select a set S subset of V such that |S boolean AND V-i| = k(i) for each i is an element of [c], maximizing the total weight of edges crossing S (i.e., edges with exactly one endpoint in S). By designing an approximate kernel for Constrained Max-Cut and building on the correlation rounding technique of Raghavendra and Tan (2012), we present a (0.858- epsilon)-approximation algorithm for the problem when c = O(1). The algorithm runs in time O min{k/epsilon, n}(poly(c/epsilon)) + poly(n), where k = Sigma(i is an element of[c]) k(i) and n = |V |. This improves upon the (1/2 + epsilon(0))-approximation of Feige and Langberg (2001) for Max-Cut(k) (the special case when c = 1, k(1) = k), and generalizes the (0.858- epsilon)approximation of Raghavendra and Tan (2012), which only applies when min{k, n - k} = Omega(n) and does not handle multiple constraints. We also establish that, for general values of c, it is NP-hard to determine whether a feasible solution exists that cuts all edges. Finally, we present a 1/2-approximation algorithm for Max-Cut under an arbitrary matroid constraint.
We show that for a randomly sampled unsatisfiable O(log n)-CNF over n variables the randomized two-party communication cost of finding a clause falsified by the given variable assignment is linear in n.
