In traditional algorithm design, no incentives come into play: the input is given, and your algorithm must produce a correct output. How much harder is it to solve the same problem when the input is not given directly, but instead reported by strategic agents with interests of their own? The unique challenge stems from the fact that the agents may choose to lie about the input in order to manipulate the behavior of the algorithm for their own interests, and tools from Game Theory are, therefore, required in order to predict how these agents will behave. We develop a new algorithmic framework with which to study such problems. Specifically, we provide a computationally efficient black-box reduction from solving any Bayesian optimization problem on “strategic input,” often called algorithmic mechanism design , to solving a perturbed version of that same optimization problem when the input is directly given, traditionally called algorithm design . We further demonstrate the power of our framework by making significant progress on several long-standing open problems. First, we provide an algorithmic extension of Myerson’s celebrated characterization of single item auctions [ 105 ] to multiple items, providing also a computationally efficient implementation of optimal auctions. Next, we design a computationally efficient 2-approximate mechanism for Bayesian job scheduling on unrelated machines, the original problem studied in Nisan and Ronen’s seminal article introducing the field of Algorithmic Mechanism Design [ 106 ]. This matches the guarantee of the best known computationally efficient algorithm when the input is directly given. Finally, leveraging a connection to our framework, we additionally prove hardness of approximation for Bayesian mechanism design with submodular bidders.
In the classical survivable network design problem (SNDP), we are given an undirected graph G = (V, E) with costs on edges and a connectivity requirement k(s, t) for each pair of vertices. The goal is to find a minimum-cost subgraph H subset of G such that every pair (s, t) is connected by k(s, t) edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS'98, Combinatorica'01] gives a 2-approximation algorithm for EC-SNDP, and a decade later, an O(k(3) log n)-approximation algorithm for VC-SNDP, where k is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS'09, Theory Comput.'12]. While there is a rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This article concerns the generalization of EC-SNDP into the subset-to-subset setting, namely Group EC-SNDP. We develop a framework, which yields the first non-trivial (true) approximation algorithm for Group EC-SNDP. Previously, only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA'15], and a true approximation algorithm is known only for the single-source variant with connectivity requirement k(S, T) is an element of {0, 1, 2} [Gupta, Krishnaswamy, and Ravi, SODA'10; Khandekar, Kortsarz, and Nutov, FSTTCS'09 and Theor. Comput. Sci.'12]. On the negative side, in terms of the number of connectivity demands q, we give an Omega (q/log q)-hardness result for large k, complementing the previous inapproximability results, e.g., hardness in terms of k: k(1/5-epsilon)- hardness [Cheriyan et al., SODA'12; Laekhanukit, SODA'14; Chalermsook et al., SODA'15; Manurangsi, IPL'19]; hardness in terms of n: 2(log1-epsilon) (n)-hardness [Chalermsook et al., SODA'15].
We develop a quantitative analogue of equational reasoning, which we call quantitative equational logic. The quantitative equations use, instead of classical equality, quantitative equalities, which are equalities indexed with nonnegative reals. Thus, s = ε t means that “ s and t are points in a metric space and their distance is less than ε”. Quantitative equalities will be used to encode behavioural distances, with ε being an upper bound on the measure of dissimilarity between two terms. We develop the metatheory of this subject. We define a notion of quantitative algebra, which is the quantitative analogue of universal algebra. We prove a completeness theorem for quantitative equational logic, and we show that we obtain monads on suitable categories of metric spaces. We present a set of examples where the free algebra of a quantitative equational theory corresponds to some well-known structure. These examples are: Hausdorff metrics from quantitative semilattices; p -Wasserstein metrics (hence also the Kantorovich metric), and the total variation metric.
An important objective function in the scheduling literature is to minimize the sum of weighted flow times. We are given a set of jobs, where each job is characterized by a release time, a processing time, and a weight. Our goal is to find a preemptive schedule on a single machine that minimizes the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its completion time and its release time. In their breakthrough result, Batra, Garg, and Kumar [FOCS 2018] found the first pseudopolynomial-time constant-factor approximation algorithm for the problem, which was turned into a polynomial-time algorithm by Feige, Kulkarni, and Li [SODA 2019]. The resulting approximation ratio is a (not explicitly stated) constant which is at least 10,000. In this article, we improve this to a PTAS. 1 The algorithm by Batra et al. reduces the problem to Demand MultiCut on trees and solves the resulting instances via LP-rounding and a dynamic program. Instead, we first reduce the problem to a (different) geometric problem while losing only a factor \(1 + \varepsilon\) , and then solve its resulting instances exactly by a dynamic program. In particular, our reduction ensures certain structural properties, due to which we do not need LP-rounding techniques.
A randomized algorithm for a search problem is pseudodeterministic if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser [16] posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an unconditional polynomial-time randomized algorithm B such that, for infinitely many values of n, B(1(n)) outputs a canonical n-bit prime p(n) with high probability. More generally, we prove that for every dense property Q of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying Q. This improves upon a subexponential-time construction of Oliveira and Santhanam [49]. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell [11], using a variant of the Shaltiel-Umans generator [51].
For any ɛ > 0, we give a simple, deterministic (4+ɛ)-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents’ valuations, and give an e(ω + 2 + ɛ)-approximation if the ratio between the largest weight and the average weight is at most ω. We also show that the 1/2-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time that is both 1/2-EFX and an (8+ɛ)-approximation to the symmetric NSW problem under submodular valuations.
The mathematical study of voting, social choice theory , has traditionally only been applicable to choices among predetermined alternatives, but not to open-ended decisions such as collectively selecting a textual statement. We introduce generative social choice , a design methodology for open-ended democratic processes that combines the rigor of social choice theory with the capability of large language models to generate text and extrapolate preferences. Our framework divides the design of AI-augmented democratic processes into two components: first, proving that the process satisfies representation guarantees when given access to oracle queries; second, empirically validating that these queries can be approximately implemented using a large language model. We apply this framework to the problem of summarizing free-form opinions into a proportionally representative set of opinion statements; specifically, we develop a democratic process with representation guarantees and use this process to portray the opinions of participants in a survey about abortion policy. In a trial with 100 representative US residents, we find that 84 out of 100 participants feel “excellently” or “exceptionally” represented by the set of five statements we extracted.
We give a new approach to the fundamental question of whether proof complexity lower bounds for concrete propositional proof systems imply super-polynomial Boolean circuit lower bounds. We observe that any general implication from proof complexity lower bounds for a propositional proof system to super-polynomial Boolean circuit lower bounds implies unconditionally that NEXP does not have Boolean circuits of polynomial size. We explore connections that are possible to establish without settling this long-standing and famously hard open question. For any poly-time computable function f, we define the witnessing formulas w(n)(k)(f), which are propositional formulas stating that for any circuit C of size n(k) on n variables and for any formula phi of size n, either C computes a satisfying assignment to q5 or f verifiably refutes that C computes SAT on instances of length n. We show that if the witnessing formulas are tautologies, then any super-polynomial lower bound for Extended Frege augmented with w(n)(k)(f) axioms implies that SAT requires super-polynomial size Boolean circuits. We also give an unconditional equivalence between circuit lower bounds for the Discrete Logarithm problem and proof complexity lower bounds (for propositional formulas efficiently encoding the statement that the Discrete Logarithm problem is computable by small circuits) for a concretely defined strong (non-uniform) propositional proof system. We give consequences of our connections for the meta-mathematics of several major questions in computational complexity, including whether one-way functions can be based on the worst-case hardness of NP, whether there is a dichotomy between one-way functions and worst-case learning with membership queries over the uniform distribution, and whether there are feasibly constructible anti-checkers for Satisfiability. We show that for each of these questions, provability of a positive answer in essentially any standard mathematical theory would imply new connections between propositional proof complexity and circuit complexity. Our results rely on a new notion of "self-provability" of upper bounds, which might be independently interesting, and involve a novel application of random self-reducibility to proof complexity.
Given a set S subset of U and a function f: S -> {0, 1} (R), a static retrieval data structure for f supports queries that return f (x) for x E S and an arbitrary value from {0, 1} r for x E U \ S. Retrieval data structures can be used to implement a static approximate membership query (AMQ) data structure, i.e., a Bloom filter alternative, with false positive rate 2(-r) . The information-theoretic space lower bound for both tasks is rSbits, and here we aim to use space r S (1 + P) bits for a small overhead P, including succinct constructions with P = o (1). A well-known approach to this task associates each key x E S with a row vector h (R)(x) is an element of {0, 1}m and stores a matrix Z is an element of {0, 1}(mxr) such that h (R)(x) & centerdot; Z = f (x) for every x E S. We propose a new variant where h (R)(x) contains a short block of random bits at a random position s (x), and is otherwise zero. Sorting the row vectors by s(x) gives a matrix A is an element of {0, 1}(n & times;m) with non-zero entries concentrated in a "ribbon" along a generalized diagonal. This makes a variant of Gaussian elimination particularly efficient at computing Z. We thus obtain simple data structures called Standard Ribbon Retrieval and Homogeneous Ribbon Filter. We then refine the construction using bumping (a variant of backyarding) and overloading (using m < n) to obtain bumped ribbon retrieval ("BuRR"), with overhead O( logw/rw(2)),query time O(1 + rw/log n), and expected construction time O(nw), for a tuning parameter w = O (log n) that opens a trade-off between space and running time. Our experiments reveal our implementations to be the first to simultaneously achieve small overheads and fast running times in practice, with BuRR achieving overheads well below 1 % while being faster than most competitors, which have larger space overheads. This efficiency, including favorable constants, stems from a combination of simplicity, word parallelism, and high locality. We offer a unified theoretical perspective on these three ribbon-based data structures, including a nontrivial rigorous analysis of their running times and memory consumption.
In 1960, Osborne proposed a simple iterative algorithm for matrix balancing with outstanding numerical performance. Today, it is the default preconditioning procedure before eigenvalue computation and other linear algebra subroutines for square non-symmetric matrices in mainstream software packages such as Python, Julia, MATLAB, EISPACK, LAPACK, and more. Despite its widespread usage, Osborne's algorithm has long resisted theoretical guarantees for its runtime: the first polynomial-time guarantees were obtained only in the past decade, and recent near-linear runtimes remain confined to variants of Osborne's algorithm with important differences that make them simpler to analyze but empirically slower. In this paper, we address this longstanding gap between theory and practice by proving that Osborne's original algorithm-the de facto matrix balancing preconditioner in practice-in fact has a near-linear runtime. This runtime guarantee (1) is optimal in the input size up to at most a single logarithm, (2) is the first runtime for Osborne's algorithm that does not dominate the runtime of downstream tasks like eigenvalue computation, and (3) improves upon the theoretical runtimes for all other variants of Osborne's algorithm.
DWe show that for all functions.. (..) =.., every multitape Turing machine running in time.. can be simulated in space only.. ( v... log..). This is a substantial improvement over Hopcroft, Paul, and Valiant's simulation of time.. in.. (.. /log..) space from 50 years ago [FOCS 1975, JACM 1977]. Among other results, our simulation implies that bounded fan-in circuits of size.. can be evaluated on any input in only v.. center dot........ (log..) space, and that there are explicit problems solvable in.. (..) space which require at least.. 2-.. time on every multitape Turing machine for all.. > 0, thereby making a little progress on the P versus PSPACE problem. Our simulation reduces the problem of simulating time-bounded multitape Turing machines to a series of implicitly-defined Tree Evaluation instances with nice parameters, leveraging the remarkable space-efficient algorithm for Tree Evaluation recently found by Cook and Mertz [STOC 2024].
The Rabin tree theorem yields an algorithm to solve the satisfiability problem for monadic second-order logic over infinite trees. Here we solve the probabilistic variant of this problem. Namely, we show how to compute the probability that a randomly chosen tree satisfies a given formula. We additionally show that this probability is an algebraic number. This closes a line of research where similar results were shown for formalisms weaker than the full monadic second-order logic.
Vizing's theorem states that any.. -vertex..-edge graph of maximum degree. can be edge colored using at most. + 1 different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in.. (....) time. This was subsequently improved to time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to by [Assadi, 2024] and by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a (.+ 1)-edge coloring in near-linear time-in fact, only.. (.. log.) time-with high probability, giving a near-optimal algorithm for this fundamental problem.
Despite the (algorithmic) importance of treewidth, both its complexity and approximability present large knowledge gaps. While the best currently known polynomial-time approximation algorithm has ratio.. (v. log OPT), no approximation factor could be ruled out under P. NP alone. There are 2.. (..) -time algorithms to compute the treewidth of..-vertex graphs, but the Exponential-Time Hypothesis (ETH) was only known to imply that 2.. (v..) time is required. The reason is that all the known hardness constructions use Cutwidth or Pathwidth on bounded-degree graphs as an intermediate step in a (long) chain of reductions, for which no inapproximability nor sharp ETH lower bound is known. We present a simple, self-contained reduction from 3-SAT to Treewidth. This starts filling the former gap, and completely fills the latter gap. Namely, we show that 1.00005-approximating Treewidth is NP-hard, and solving Treewidth exactly requires 2.. (..) time, unless the ETH fails. We further derive, under the latter assumption, that there are some constants.. > 1 and.. > 0 such that.. -approximating Treewidth requires time 2.. (../log....).
A semi-streaming algorithm in dynamic graph streams processes any n-vertex graph by making one or multiple passes over a stream of insertions and deletions to edges of the graph and using O(n ·(n)) space. Semi-streaming algorithms for dynamic streams were first obtained in the seminal work of Ahn, Guha, and McGregor in 2012, alongside the introduction of the graph sketching technique, which remains the de facto way of designing algorithms in this model and a highly popular technique for designing graph algorithms in general. We settle the pass complexity of approximating maximum matchings in dynamic streams via semi-streaming algorithms by improving the state-of-the-art in both upper and lower bounds. We present a randomized sketching based semi-streaming algorithm for O(1)-approximation of maximum matching in dynamic streams using O(loglogn) passes. The approximation ratio of this algorithm can be improved to (1+ϵ) for any fixed ϵ > 0 even on weighted graphs using standard techniques. This exponentially improves upon several O(logn) pass algorithms developed for this problem since the introduction of the dynamic graph streaming model. In addition, we prove that any semi-streaming algorithm (not only sketching based) for O(1)-approximation of maximum matching in dynamic streams requires Ω(loglogn) passes. This presents the first multi-pass lower bound for this problem, which is already also optimal, settling a longstanding open question in this area.
We show that there is a language in \(\textsf{S}_2\textsf {E}\) (symmetric exponential time) that requires circuit complexity at least \(2^n/n\) on every input length. In particular, the above also implies the same near-maximum circuit lower bounds for \(\Sigma _2\textsf {E}\cap \Pi _2\textsf {E}\) and \(\mathsf {ZPE}^{\textsf {NP}}\) . Our proofs relativise. Previously, only “half-exponential” circuit lower bounds for the aforementioned complexity classes were known, and the smallest complexity class known to require exponential circuit complexity was \(\Delta _3\textsf {E}= \textsf {E}^{\Sigma _2\textsf{P}}\) (Miltersen, Vinodchandran, and Watanabe COCOON’99). Our circuit lower bounds are corollaries of an unconditional zero-error pseudodeterministic algorithm with an \(\textsf {NP}\) oracle that solves the Range Avoidance problem. This algorithm also implies unconditional pseudodeterministic \(\textsf {FZPP}^{\textsf {NP}}\) constructions for Ramsey graphs, rigid matrices, two-source extractors, linear codes, and \(\mathrm{K}^{\mathrm{poly}}\) -random strings with nearly optimal parameters.
We propose a semantic foundation for logics for reasoning in settings that possess a distinction between equality of variables, a coarser equivalence of variables, and a notion of conditional independence between variables.We show that such relations can be modelled naturally in atomic sheaf toposes.Equivalence of variables is modelled by an intrinsic relation of atomic equivalence that is possessed by every atomic sheaf.We identify additional structure on the category generating the atomic topos (primarily, the existence of a system of independent pullbacks) that allows conditional independence to be interpreted in the topos.We then study the logic of equivalence and conditional independence that is induced by the internal logic of the topos.This atomic sheaf logic is a classical logic that validates a number of fundamental reasoning principles relating equivalence and conditional independence.As a concrete example of this abstract framework, we use the atomic topos over the category of surjections between finite nonempty sets as our main running example.In this category, the interpretations of equivalence and conditional independence coincide with those given by the multiteam semantics of independence logic, in which the role of equivalence is taken by the relation of mutual inclusion.A major difference from independence logic is that, in atomic sheaf logic, the multiteam semantics of the equivalence and conditional independence relations is embedded within a classical surrounding logic.At the end of the paper, we briefly outline two other instances of our framework, to demonstrate its versatility.The first of these is a category of probability sheaves, in which atomic equivalence is equality-in-distribution, and the conditional independence relation is the usual probabilistic one.Our other example is the Schanuel topos (equivalent to nominal sets) where equivalence is orbit equality and conditional independence amounts to a relative form of separatedness.
We show that for all epsilon > 0, for sufficiently large prime power q epsilon N, for all delta > 0, it is NP-hard to distinguish whether a 2-Prover1-Round projection game with alphabet size q has value at least 1 - delta, or value at most 1/q(1-epsilon). This establishes a nearly optimal alphabet-to-soundness tradeoff for 2-query PCPs with alphabet size q, improving upon a result of [Chan 2016]. Our result has the following implications: (1) Near optimal hardness for Quadratic Programming: it is NP-hard to approximate the value of a given Boolean Quadratic Program within factor (log n)(1-o(1)) under quasi-polynomial time reductions. This result improves a result of [Khot-Safra 2013] and nearly matches the performance of the best known approximation algorithm [Megrestki 2001, Nemirovski-Roos-Terlaky 1999 Charikar-Wirth 2004] that achieves a factor of O (log n). (2) Bounded degree 2-CSP's: under randomized reductions, for sufficiently large d > 0, it is NP-hard to approximate the value of 2-CSPs in which each variable appears in at most 3 constraints within factor (1 - o(1)) d/2, improving upon a recent result of [Lee-Manurangsi 2023]. (3) Improved hardness results for connectivity problems: using results of [Laekhanukit 2014] and [Manurangsi 2019], we deduce improved hardness results for the Rooted k-Connectivity Problem, the Vertex-Connectivity Survivable Network Design Problem and the Vertex-Connectivity k-Route Cut Problem.
