Full-information protocols are a common abstraction in distributed computing. In its iterated form, processes proceed through successive communication rounds in which each process shares its entire current knowledge of the execution. Usually, it is implicitly assumed that a process can transmit an unbounded amount of information in every round. This paper makes that assumption explicit and studies its computational cost.We formalize the problem of simulating iterated full-information protocols when communication is bounded, so that each process may share at most b bits with every other process per round. We show that for a large family of shared-memory communication models and n+1 processes, simulating r iterations of the full-information protocol requires Ω((n!)r−12n−b) rounds. We complement this lower bound with simulation algorithms that are asymptotically optimal in the conventional read-write model. In particular, the case of two processes is special: r iterations of a full-information protocol can be simulated in exactly r rounds while using only 2-bit variables, as the amount of information that must be represented remains bounded for two processes but grows with each iteration for n ≥ 2.
So Long Sucker is a strategy board game developed by Mel Hausner, John Nash, Lloyd Shapley, and Martin Shubik in 1964. It requires 4 players, each with c chips of their designated color, and a board made of k empty piles. With a clear set-up comes intricate rules, such as: players taking turns but not in a fixed order, agreements made between some players broken at any time, or a player winning the game without any chips in hand.One of the main points of interest in studying this game is finding when a player has a winning strategy. The game begins with four players who get successively eliminated until only the winner is left. To study winning strategies, it is of interest to look at endgame situations. For that, we study the following game set-up: there are two players left in the game, Blue and Red, with only their respective chip colors. In this paper, we present a formal description of the rules of the game and characterize Blue’s winning scenarios and strategies for this game set-up through a delicate case analysis. To the best of our knowledge, these are the first theoretical results about this fundamental game.
Selecting a representative subset of tuples from a database to support multi-criteria decision-making is an important problem in big data processing. A popular method for this problem is the happiness maximization query (HMQ) (a.k.a. regret minimization query or RMQ), which returns a set of k tuples that make users happy (a.k.a. not regretful) in the sense that all of them can find at least one tuple in the selected subset that is not much worse than their favorites in the database. Although extensive studies have been conducted on HMQ, most of them cannot work efficiently in an online setting over streaming data, where tuples arrive sequentially and an algorithm must maintain a query result in real time according to the observed tuples. In this paper, we investigate the online variant of HMQ (Online-HMQ) over data streams. Specifically, we measure user satisfaction as the minimum happiness ratio across all users. We show that the problem is NP-hard in databases of three or higher dimensions. We prove that the objective function of Online-HMQ is monotone but non-submodular. We subsequently convert it into a submodular function using the truncation technique. As such, we propose two online algorithms for submodular maximization, ThresholdOnline and SieveOnline, to process Online-HMQ with theoretical guarantees. Comprehensive experiments on synthetic and real-world datasets confirm that our proposed Online-HMQ algorithms are effective, efficient, and scalable.
We construct a lattice-based identity-based signature (IBS) scheme that achieves existential unforgeability under chosen-identity and chosen-message attacks (EUF-ID-CMA) in the standard model. Note that existing IBS schemes from lattices are either based on the random oracle model, only achieve selective security in the standard model, or only have a variant EUF-ID-CMA (denoted EUF-ID-CMA‾) security. The EUF-ID-CMA‾ security model requires the user to obtain a fresh signing key for signing each message, and thus it can hardly capture real-world applications. Therefore, our IBS scheme serves as the first EUF-ID-CMA secure IBS from lattices in the standard model.We also propose a new framework of constructing identity-based matchmaking encryption (IB-ME) from IBS and identity-based encryption (IBE). We modify the stronger authenticity for IB-ME (Wang et al. TCS 2025) to a new stronger authenticity. We prove that the new stronger authenticity of IB-ME can be reduced to the EUF-ID-CMA security of the underlying IBS and the ciphertext pseudo-randomness of IB-ME can be reduced to the ciphertext pseudo-randomness of the underlying IBE. Our lattice-based IBS scheme and the IBE scheme (Boyen and Li AsiaCrypt 2016) yield the first lattice-based IB-ME scheme with new stronger authenticity in the standard model.
Given a bipartite graph G = (T ∪ B, E) and a positive integer k, the NP-complete bipartite one-sided vertex explosion problem asks whether G admits a planar 2-layer drawing after exploding at most k vertices of B. Parameterized algorithms for this problem have received increasing attention more recently. Ahmed et al. (GD2022) gave a kernel of O(k6) vertices and a parameterized algorithm with running time in 2O(k6)·m, where m is the number of edges of G. Subsequently, Baumann et al. (WG2023) more generally studied the problem of asking whether a graph G = (V, E) has pathwidth at most 1 after exploding at most k vertices in a given subset of V, which was shown to admit a kernel of at most 16k2+16k vertices and a search-tree algorithm with running time in O(4k·m). In this paper, we continue the study of bipartite one-sided vertex explosion parameterized by k. Our main contribution is to show that this parameterized problem admits a kernel of at most 10k vertices, and admits a search-tree algorithm with running time in O(2.31k·m).
Let P be the set of prime numbers. We show that for each finite set Π⊆P∖{2}, there exists a subring ΛΠ⊆Q and an injective polynomial function P: ΛΠ × ΛΠ → ΛΠ.
Fo-bicategories can be seen as a categorical version of Peirce’s calculus of relations. What makes them particularly interesting is that their rules form a complete system for first-order logic using only equations–no extra logical machinery is needed. In this paper, we show how fo-bicategories relate to Lawvere’s hyperdoctrines. To make our main result easier to prove and understand, we introduce a simpler structure called peircean bicategories, which capture the essence of fo-bicategories in a more straightforward way.
Scheduling within a limited budget is closely related to several much studied scheduling with compression and rescheduling problems, and is also a variant of the more recent model of scheduling with testing. In this problem, one seeks to minimize the makespan for a set of jobs that are to be processed on a number of parallel identical machines. Each job Jj is given an upper bound uj on its actual processing time pj, and a testing fee cj. In the offline case, the processing time pj is known to the scheduler; while in the oblivious case, pj is revealed to the scheduler only if the testing fee cj is paid. So the scheduler can choose to execute Jj on any one of the machines non-preemptively for uj time or pay for the testing and then execute the job for pj time. The scheduler is given a budget B to pay for testing and seeks to minimize the makespan within that budget. The offline problem is denoted as P∣uj, pj, cj, B∣Cmax, and the oblivious problem is denoted as P∣uj,−,cj,B∣Cmax, where P stands for multiple parallel identical machines with the number of machines being part of the input. We contribute a polynomial-time approximation scheme (PTAS) for the offline problem P∣uj, pj, cj, B∣Cmax, which leads to an almost tight (2+ϵ)-competitive algorithm for the oblivious problem P∣uj,−,cj,B∣Cmax.
A set of intervals I={I1,I2,⋯,In} forms a simple chain if, for every 2≤i≤n−1, interval Ii overlaps only with Ii−1 and Ii+1. We show that a deterministic memoryless one-directional revoking algorithm achieves a competitive ratio of 2(1−1/e)≈0.786 on the simple chain in the random order model, hence performs worse than the basic greedy algorithm without revoking that has a competitive ratio of (1−1/e2)≈0.864, but better than any deterministic revoking algorithm in the adversarial model that has a competitive ratio of at most 0.75. The proof of the latter also leads to a lower bound of n/4 for the advice complexity.
In the era of network science and big data, fortifying the resilience of multiprocessor systems has emerged as a critical concern. Consequently, to comprehensively and rigorously assess the robustness of such systems, researchers have introduced various extensions and generalizations of the traditional concept of connectivity in graph theory. Among these, dual-constraint connectivity stands out as a pivotal metric for evaluating both the robustness and fault-tolerance capabilities of interconnection networks. Formally, for a given interconnection network represented as a graph G and two positive integers h and r, the h-extra r-component connectivity of G, designated as cκrh(G), is defined as the cardinality of the smallest node set F (i.e., the minimum number of nodes) whose removal results in the graph G−F being partitioned into no fewer than r connected components and each component contains at least h+1 nodes. In this paper, we present a comprehensive theoretical analysis of h-extra r-component connectivity for a class of hypercube-based compound architectures. Our study encompasses a wide range of network topologies, including hypercubes, bicube networks, hierarchical cubic networks, half hypercubes, folded hypercubes, and hierarchical hypercubes. Through rigorous mathematical derivations and proofs, we establish general theoretical characterizations of this connectivity metric, enhancing the fault tolerance and reliability of complex interconnection networks.
Ambainis and de Wolf proposed the concept of average-case query complexity of boolean functions and showed that average-case deterministic, average-case bounded-error randomized, and average-case quantum query complexities are not polynomially related. While tools such as the OSSS/OS inequalities provide lower bounds on the average-case deterministic query complexity under a uniform distribution, denoted Dave(f), the understanding of upper bounds is much more limited. To address this gap, we prove two upper bounds on Dave(f) and demonstrate their tightness.First, we establish an upper bound on Dave(f) in terms of the weight wt(f), i.e., the number of inputs on which the output is 1. Specifically, for every f: {0, 1}n → {0, 1} with wt(f)≥4logn, Dave(f)≤logwt(f)logn+O(loglogwt(f)logn), and Dave(f)=O(1) when wt(f)<4logn. Moreover, we show that the upper bound is tight up to an additive logarithmic term for almost all boolean functions.Second, we prove that Dave(F)≤n(1−lognO(k)) for any k-CNF F and log n ≤ k ≤ n0.99, by adapting techniques from Håstad’s switching lemma proof and extending the worst-case analysis to the average-case setting. Moreover, we show that the above bound is tight by exhibiting a k-CNF F with Dave(F)=n(1−lognΘ(k)) for any k ≥ 2log n.
Combinatorial Optimization (CO) has made significant strides in accuracy and computational efficiency through the adoption of Machine Learning (ML) techniques. However, previous studies have not fully addressed the symmetries inherent in CO. This paper introduces a novel training approach, SA-NS, which employs a regularizer-based method to exploit universal symmetries present in various CO problems and their solutions. By leveraging symmetry alignment, this approach substantially enhances the generalization capability of neural heuristic solver. It enables learned solvers to effectively utilize common symmetries within the same class of CO problems. Our experiments demonstrate that SA-NS significantly enhance the performance of neural solver methods in two CO problems—the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP)—all without relying on problem-specific domain expertise.
Inspired by the FLIP stream cipher introduced by Méaux et al. at Eurocrypt 2016, Boolean functions with good cryptographic properties when their inputs are limited to specific subsets of F2n are well studied. Weightwise perfectly balanced (WPB) functions are a class of Boolean functions that remain balanced across all subsets of F2n with a fixed Hamming weight ranging from 1 to n−1. This paper introduces novel constructions of WPB functions by adjusting the supports of a family of Boolean functions with degree 8. The resulting WPB functions have high algebraic degree, weightwise nonlinearity and algebraic immunity.
A suffixient set is a novel combinatorial object that captures the essential information of repetitive strings in a way that, provided with a random access mechanism, supports various forms of pattern matching. In this paper, we study the size χ of the smallest suffixient set as a repetitiveness measure.First, we study its sensitivity to various string operations. We show that χ cannot increase by more than 2 after appending or prepending a character to the string. As a consequence, we are able to give simple linear-time online algorithms to compute smallest suffixient sets. We also show that, although reversing the string can increase χ by an arbitrary O(n) value, it always holds χ(T)/χ(TR) ≤ 2. We also prove lower and upper bounds for the additive or multiplicative increase of χ after applying arbitrary edit operations, or rotating the text. In particular, we show that the additive increase can be as large as Ω(n) for all those operations.Secondly, we place χ among known repetitiveness measures. In particular, we show χ ≤ 2r (where r is the number of runs in the Burrows-Wheeler Transform of the string), that there are string families where χ=o(v) (where v is the size of the smallext lexicographic parse of the string), and that χ is uncomparable to almost all reachable measures based on copy-paste mechanisms. In passing, we give precise bounds for χ for some relevant string families, for example χ≤σ+2 on episturmian words over alphabets of size σ (e.g., χ ≤ 4 on Fibonacci strings, for which we precisely characterize the only two smallest suffixient sets).
Virus machines (VMs) provide a computing paradigm inspired by the transmission and replication networks of viruses. The model consists of processing units, called hosts, connected by a directed graph of channels, alongside an instruction graph that governs the transmission of virus objects between hosts. To further delineate the computational power of this model, this paper introduces a series of normal forms that impose structural and operational restrictions on VMs. Specifically, we investigate bounded variants that restrict the number of hosts, instructions, and virus objects per host, as well as the size of network loops. By establishing these normal forms, we prove new characterizations for fundamental families of number sets -namely finite, semilinear, and recursively enumerable (NRE) sets- thereby refining the computability landscape of virus machines.
Convex analysis and Gaussian probability are tightly connected, as mostly evident in the theory of linear regression. Our work introduces an algebraic perspective on this relationship, in the form of a diagrammatic calculus of string diagrams, called Graphical Quadratic Algebra (GQA). We show that GQA is a complete axiomatisation for the category of quadratic relations, a compositional formulation of quadratic optimisation problems. Moreover, we identify a sub-theory of GQA which is complete for the category of Gaussian probabilistic processes. We show how GQA may be used to study linear regression, probabilistic programming, and electrical circuits.
