Bence Bakos, Norbert Hegyvári, Máté Pálfy
The original knapsack problem is well known to be NP-complete. In a multidimensional version one have to decide whether a $p\in \N^k$ is in a sumset-sum of a set $X \subseteq \N^k$ or not. In this paper we are going to investigate a communication complexity problem related to this.
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Kolmogorov complexity as a combinatorial tool
本研究提出Kolmogorov复杂度可作为精妙的组合工具，给出了用其证明自然组合博弈必胜策略存在性的意外示例，补充了该复杂度理论在组合领域的应用场景。
Forbidden substrings, Kolmogorov complexity and almost periodic sequences
假设对于某些$α<1$和对于所有自然$n$，最多$2^{αn}$“禁止的”长度为$n$的二进制串的集合$F_n$是固定的。然后存在一个无限二进制序列$ω$，它没有（长）禁止子串。我们证明了这一组合语句翻译成一个关于Kolmogorov复杂性的声明，并比较这个证明与组合的基础上Laslo Lovasz局部引理。然后我们构造了一个具有相同性质的概周期序列（从而结合了Levin和Muchnik-Semenov-Ushakov的结果）.组合证明和Kolmogorov复杂性论证都可以推广到多维情况。
Optical orthogonal codes from a combinatorial perspective
该研究从组合视角出发，解耦光正交码的自相关与互相关值，研究相关值大于1的情形，分析相关界、结构意义及与其他组合对象的关联，梳理已有成果并提出新概念与开放问题。
Combinatorial Characterization of Formal Languages
本文是作者在物理学和数学博士学位（在俄罗斯）的论文的扩展摘要。本文研究的主要特征是组合复杂性，它是一种与语言相关的“计数”函数，返回该语言中给定长度的单词的数量。对于几类语言，研究了关于组合复杂性及其与语言其他参数的关系的各种问题。简要介绍的主题和配方的结果。没有证据，而是引用了包含证据的论文。
A novel polynomial expansion method of symmetric Boolean functions is described. The method is efficient for symmetric Boolean function with small set of valued numbers and has the linear complexity for elementary symmetric Boolean functions, while the complexity of the known methods for this class of functions is quadratic. The proposed method is based on the consequence of the combinatorial Lucas theorem.
In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with $n$ nodes. The best known lower bound is the trival (dimension) bound, $Ω(n^2)$, the best known upper bound is the extended formulation by Wong (1980) of size $O(n^3)$ (also Martin, 1991). In this note we give a nondeterministic communication protocol with cost $\log_2(n^2\log n)+O(1)$ for the support of the spanning tree slack matrix. This means that the combinatorial lo…
Unique Sink Orientations (USOs) are an appealing abstraction of several major optimization problems of applied mathematics such as for instance Linear Programming (LP), Markov Decision Processes (MDPs) or 2-player Turn Based Stochastic Games (2TBSGs). A polynomial time algorithm to find the sink of a USO would translate into a strongly polynomial time algorithm to solve the aforementioned problems---a major quest for all three cases. In addition, we may translate MDPs and 2TBSGs into the problem of finding the sink of an acyclic USO of a cube, which can be done using the well-known Policy Iter…
Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Qua…
A long-standing conjecture by Heath, Pemmaraju, and Trenk states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are $at$-outerplanar graphs. On the complexity side, it is known that deciding whether a graph has upward book thickness $k$ is NP-hard for any fixed $k \ge 3$. We show that the problem, for any $k \ge 5$, remains NP-hard for gra…
We refute the Probabilistic Universal Graph Conjecture of Harms, Wild, and Zamaraev, which states that a hereditary graph property admits a constant-size probabilistic universal graph if and only if it is stable and has at most factorial speed. Our counter-example follows from the existence of a sequence of $n \times n$ Boolean matrices $M_n$, such that their public-coin randomized communication complexity tends to infinity, while the randomized communication complexity of every $\sqrt{n}\times \sqrt{n}$ submatrix of $M_n$ is bounded by a universal constant.
On the Computational Complexity of Multi-Objective Ordinal Unconstrained Combinatorial Optimization
针对多目标无约束组合优化问题（MUCO）通常难求解的问题，本文探索其易解特殊情况，证明至多两序数加一实值目标的MUCO问题可多项式时间求解并计算完整非支配集，还为单序数加另一序数/实值目标的情况提出更高效的多贪心策略算法。
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices", held in February 2013 at Dagstuhl Castle.
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory
组合博弈在算法和复杂性理论中引出了几个有趣的、清晰的问题，其中许多问题仍然是开放的。本文的目的是提供该领域的概述，以鼓励进一步的研究。特别是，我们开始在组合博弈论的一般背景，它分析了理想的发挥在完美的信息游戏，和约束逻辑，它提供了一个框架，显示硬度。然后，我们调查结果的复杂性，确定理想的发挥在这些游戏中，解决难题的相关问题，在多项式时间算法和计算的棘手性的结果。我们对背景的回顾和对算法结果的调查绝不是完整的，但应该作为一个有用的入门。
The modern integrated circuit is one of the most complex products that has been engineered to-date. It continues to grow in complexity as the years progress. As a result, very large-scale integrated (VLSI) circuit design now involves massive design teams employing state-of-the art computer-aided design (CAD) tools. One of the oldest, yet most important CAD problems for VLSI circuits is physical design automation, where one needs to compute the best physical layout of millions to billions of circuit components on a tiny silicon surface \cite{Lim08}. The process of mapping an electronic design t…
We consider the learning and communication complexity of subsequence containment. In the learning problem, we seek to learn a classifier that positively labels a binary string $x$ if it contains a fixed binary string $y$ as a subsequence. In the communication problem, $x$ and $y$ are partitioned between two players, Alice and Bob, who wish to determine if $x$ contains $y$ as a subsequence using a minimal amount of communication. We devise asymptotically tight bounds for the sample complexity (VC dimension) of the learning problem and the communication complexity of the communication problem. O…
Suppose that Alice and Bob are given each an infinite string, and they want to decide whether their two strings are in a given relation. How much communication do they need? How can communication be even defined and measured for infinite strings? In this article, we propose a formalism for a notion of infinite communication complexity, prove that it satisfies some natural properties and coincides, for relevant applications, with the classical notion of amortized communication complexity. More-over, an application is given for tackling some conjecture about tilings and multidimensional sofic sh…
The log-rank conjecture in communication complexity suggests that the deterministic communication complexity of any Boolean rank-r function is bounded by polylog(r). Recently, major progress was made by Lovett who proved that the communication complexity is bounded by O(r^1/2 * log r). Lovett's proof is based on known estimates on the discrepancy of low-rank matrices. We give a simple, direct proof based on a hyperplane rounding argument that in our opinion sheds more light on the reason why a root factor suffices and what is necessary to improve on this factor.
An archetypal problem discussed in computer science is the problem of searching for a given number in a given set of numbers. Other than sequential search, the classic solution is to sort the list of numbers and then apply binary search. The binary search problem has a complexity of O(logN) for a list of N numbers while the sorting problem cannot be better than O(N) on any sequential computer following the usual assumptions. Whenever the problem of deciding partial order can be done in O(1), a variation of the problem on some bounded list of numbers is to apply binary search without resorting …
VC v. VCG: Inapproximability of Combinatorial Auctions via Generalizations of the VC Dimension
具有适当近似比的组合拍卖激励相容计算有效协议的存在性是计算机制设计中的典型问题。据信，在许多情况下，很好的近似组合拍卖可能是无法实现的，由于真实性和计算效率之间的固有冲突。然而，到目前为止，研究人员缺乏证明这些结果的机制。在本文中，我们提出了一种新的方法，我们相信有很大的希望在这个重要的问题上取得进展。我们迈出了开发新技术的第一步，以降低不相交集合的k元组的VC维度。我们应用这种机制来证明第一个计算复杂性的不可近似性结果的激励相容机制的组合拍卖。这些结果适用于一类重要的基于VCG的机制，并基于NP没有多项式大小的电路的复杂性假设。
We study nondeterministic communication complexity and related concepts (fooling sets, fractional covering number) of random functions $f\colon X\times Y \to \{0,1\}$ where each value is chosen to be 1 independently with probability $p=p(n)$, $n := |X|=|Y|$.
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