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  • Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess is Hard
    arXiv.cs.CC Pub Date : 2020-10-19
    Josh Brunner; Erik D. Demaine; Dylan Hendrickson; Julian Wellman

    We prove PSPACE-completeness of two classic types of Chess problems when generalized to n-by-n boards. A "retrograde" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is "valid" or "legal" or "reachable". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question

    更新日期:2020-10-20
  • Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs
    arXiv.cs.CC Pub Date : 2020-10-19
    Sebastian Berndt; Klaus Jansen; Alexandra Lassota

    We consider ILPs, where each variable corresponds to an integral point within a polytope $\mathcal{P}$, i. e., ILPs of the form $\min\{c^{\top}x\mid \sum_{p\in\mathcal P\cap \mathbb Z^d} x_p p = b, x\in\mathbb Z^{|\mathcal P\cap \mathbb Z^d|}_{\ge 0}\}$. The distance between an optimal fractional solution and an optimal integral solution (called proximity) is an important measure. A classical result

    更新日期:2020-10-20
  • Log-rank and lifting for AND-functions
    arXiv.cs.CC Pub Date : 2020-10-18
    Shachar Lovett; Alexander Knop; Sam McGuire; Weiqiang Yuan

    Let $f: \{0,1\}^n \to \{0, 1\}$ be a boolean function, and let $f_\land (x, y) = f(x \land y)$ denote the AND-function of $f$, where $x \land y$ denotes bit-wise AND. We study the deterministic communication complexity of $f_\land$ and show that, up to a $\log n$ factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of $f_\land$. This comes within a $\log

    更新日期:2020-10-20
  • Mad Science is Provably Hard: Puzzles in Hearthstone's Boomsday Lab are NP-hard
    arXiv.cs.CC Pub Date : 2020-10-17
    Michael Hoffmann; Jayson Lynch; Andrew Winslow

    We consider the computational complexity of winning this turn (mate-in-1 or "finding lethal") in Hearthstone as well as several other single turn puzzle types introduced in the Boomsday Lab expansion. We consider three natural generalizations of Hearthstone (in which hand size, board size, and deck size scale) and prove the various puzzle types in each generalization NP-hard.

    更新日期:2020-10-20
  • On the Hardness of Average-case k-SUM
    arXiv.cs.CC Pub Date : 2020-10-17
    Zvika Brakerski; Noah Stephens-Davidowitz; Vinod Vaikuntanathan

    In this work, we show the first worst-case to average-case reduction for the classical $k$-SUM problem. A $k$-SUM instance is a collection of $m$ integers, and the goal of the $k$-SUM problem is to find a subset of $k$ elements that sums to $0$. In the average-case version, the $m$ elements are chosen uniformly at random from some interval $[-u,u]$. We consider the total setting where $m$ is sufficiently

    更新日期:2020-10-20
  • Solving Shisen-Sho boards
    arXiv.cs.CC Pub Date : 2020-10-18
    Michiel de Bondt

    We give a simple proof of that determining solvability of Shisen-Sho boards is NP-complete. Furthermore, we show that under realistic assumptions, one can compute in logarithmic time if two tiles form a playable pair. We combine an implementation of the algoritm to test playability of pairs with my earlier algorithm to solve Mahjong Solitaire boards with peeking, to obtain an algorithm to solve Shisen-Sho

    更新日期:2020-10-20
  • Improving Schroeppel and Shamir's Algorithm for Subset Sum via Orthogonal Vectors
    arXiv.cs.CC Pub Date : 2020-10-16
    Jesper Nederlof; Karol Węgrzycki

    We present an $\mathcal{O}^\star(2^{0.5n})$ time and $\mathcal{O}^\star(2^{0.249999n})$ space randomized algorithm for solving worst-case Subset Sum instances with $n$ integers. This is the first improvement over the long-standing $\mathcal{O}^\star(2^{n/2})$ time and $\mathcal{O}^\star(2^{n/4})$ space algorithm due to Schroeppel and Shamir (FOCS 1979). We breach this gap in two steps: (1) We present

    更新日期:2020-10-20
  • Barrington Plays Cards: The Complexity of Card-based Protocols
    arXiv.cs.CC Pub Date : 2020-10-16
    Pavel Dvořák; Michal Koucký

    In this paper we study the computational complexity of functions that have efficient card-based protocols. Card-based protocols were proposed by den Boer [EUROCRYPT '89] as a means for secure two-party computation. Our contribution is two-fold: We classify a large class of protocols with respect to the computational complexity of functions they compute, and we propose other encodings of inputs which

    更新日期:2020-10-19
  • Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles
    arXiv.cs.CC Pub Date : 2020-10-16
    Suman K. Bera; Noujan Pashanasangi; C. Seshadhri

    Counting homomorphisms of a constant sized pattern graph $H$ in an input graph $G$ is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input $G$ and the pattern $H$. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus

    更新日期:2020-10-19
  • Complexity Measures on the Symmetric Group and Beyond
    arXiv.cs.CC Pub Date : 2020-10-14
    Neta Dafni; Yuval Filmus; Noam Lifshitz; Nathan Lindzey; Marc Vinyals

    We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity

    更新日期:2020-10-17
  • Large Very Dense Subgraphs in a Stream of Edges
    arXiv.cs.CC Pub Date : 2020-10-15
    Claire Mathieu; Michel de Rougemont

    We study the detection and the reconstruction of a large very dense subgraph in a social graph with $n$ nodes and $m$ edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log n)$. A subgraph $S$ is very dense if it has $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$. Our detection algorithm

    更新日期:2020-10-17
  • Optimal Low-Degree Hardness of Maximum Independent Set
    arXiv.cs.CC Pub Date : 2020-10-13
    Alexander S. Wein

    We study the algorithmic task of finding a large independent set in a sparse Erd\H{o}s-R\'{e}nyi random graph with $n$ vertices and average degree $d$. The maximum independent set is known to have size $(2 \log d / d)n$ in the double limit $n \to \infty$ followed by $d \to \infty$, but the best known polynomial-time algorithms can only find an independent set of half-optimal size $(\log d / d)n$. We

    更新日期:2020-10-14
  • On the Complexity of Some Facet-Defining Inequalities of the QAP-polytope
    arXiv.cs.CC Pub Date : 2020-10-13
    Pawan Aurora; Hans Raj Tiwary

    The Quadratic Assignment Problem (QAP) is a well-known NP-hard problem that is equivalent to optimizing a linear objective function over the QAP polytope. The QAP polytope with parameter $n$ - \qappolytope{n} - is defined as the convex hull of rank-$1$ matrices $xx^T$ with $x$ as the vectorized $n\times n$ permutation matrices. In this paper we consider all the known exponential-sized families of facet-defining

    更新日期:2020-10-14
  • Digraph Coloring and Distance to Acyclicity
    arXiv.cs.CC Pub Date : 2020-10-13
    Ararat Harutyunyan; Michael Lampis; Nikolaos Melissinos

    In $k$-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most $k$ sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) $k$-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question what happens when the input is "almost" acyclic. In this paper we study

    更新日期:2020-10-14
  • Trace Reconstruction Problems in Computational Biology
    arXiv.cs.CC Pub Date : 2020-10-12
    Vinnu Bhardwaj; Pavel A. Pevzner; Cyrus Rashtchian; Yana Safonova

    The problem of reconstructing a string from its error-prone copies, the trace reconstruction problem, was introduced by Vladimir Levenshtein two decades ago. While there has been considerable theoretical work on trace reconstruction, practical solutions have only recently started to emerge in the context of two rapidly developing research areas: immunogenomics and DNA data storage. In immunogenomics

    更新日期:2020-10-14
  • Counting Subgraphs in Degenerate Graphs
    arXiv.cs.CC Pub Date : 2020-10-12
    Lior Gishboliner; Yevgeny Levanzov; Asaf Shapira

    We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well

    更新日期:2020-10-14
  • Nearly Optimal Average-Case Complexity of Counting Bicliques Under SETH
    arXiv.cs.CC Pub Date : 2020-10-12
    Shuichi Hirahara; Nobutaka Shimizu

    In this paper, we seek a natural problem and a natural distribution of instances such that any $O(n^{c-\epsilon})$-time algorithm fails to solve most instances drawn from the distribution, while the problem admits an $n^{c+o(1)}$-time algorithm that correctly solves all instances. Specifically, we consider the $K_{a,b}$ counting problem in a random bipartite graph, where $K_{a,b}$ is a complete bipartite

    更新日期:2020-10-13
  • On lattice point counting in $Δ$-modular polyhedra
    arXiv.cs.CC Pub Date : 2020-10-12
    D. V. Gribanov; N. Yu. Zolotykh

    Let a polyhedron $P$ be defined by one of the following ways: (i) $P = \{x \in R^n \colon A x \leq b\}$, where $A \in Z^{(n+k) \times n}$, $b \in Z^{(n+k)}$ and $rank\, A = n$; (ii) $P = \{x \in R_+^n \colon A x = b\}$, where $A \in Z^{k \times n}$, $b \in Z^{k}$ and $rank\, A = k$. And let all rank order minors of $A$ be bounded by $\Delta$ in absolute values. We show that the short rational generating

    更新日期:2020-10-13
  • A Lower Bound for Polynomial Calculus with Extension Rule
    arXiv.cs.CC Pub Date : 2020-10-12
    Yaroslav Alekseev

    In this paper we study an extension of the Polynomial Calculus proof system where we can introduce new variables and take a square root. We prove that an instance of the subset-sum principle, the binary value principle, requires refutations of exponential bit size over rationals in this system. Part and Tzameret proved an exponential lower bound on the size of Res-Lin (Resolution over linear equations)

    更新日期:2020-10-13
  • FPRAS via MCMC where it mixes torpidly (and very little effort)
    arXiv.cs.CC Pub Date : 2020-10-12
    Jin-Yi Cai; Tianyu Liu

    Is Fully Polynomial-time Randomized Approximation Scheme (FPRAS) for a problem via an MCMC algorithm possible when it is known that rapid mixing provably fails? We introduce several weight-preserving maps for the eight-vertex model on planar and on bipartite graphs, respectively. Some are one-to-one, while others are holographic which map superpositions of exponentially many states from one setting

    更新日期:2020-10-13
  • A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Privacy
    arXiv.cs.CC Pub Date : 2020-10-10
    Marcel Dall'Agnol; Tom Gur; Oded Lachish

    We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes $q$ adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with $n^{1- 1/O(q^2 \log^2 q)}$ sample complexity, following the definition of Goldreich

    更新日期:2020-10-13
  • A Refined Laser Method and Faster Matrix Multiplication
    arXiv.cs.CC Pub Date : 2020-10-12
    Josh Alman; Virginia Vassilevska Williams

    The complexity of matrix multiplication is measured in terms of $\omega$, the smallest real number such that two $n\times n$ matrices can be multiplied using $O(n^{\omega+\epsilon})$ field operations for all $\epsilon>0$; the best bound until now is $\omega<2.37287$ [Le Gall'14]. All bounds on $\omega$ since 1986 have been obtained using the so-called laser method, a way to lower-bound the `value'

    更新日期:2020-10-13
  • The Complexity of Counting Edge Colorings for Simple Graphs
    arXiv.cs.CC Pub Date : 2020-10-10
    Jin-Yi Cai; Artem Govorov

    We prove #P-completeness results for counting edge colorings on simple graphs. These strengthen the corresponding results on multigraphs from [4]. We prove that for any $\kappa \ge r \ge 3$ counting $\kappa$-edge colorings on $r$-regular simple graphs is #P-complete. Furthermore, we show that for planar $r$-regular simple graphs where $r \in \{3, 4, 5\}$ counting edge colorings with \k{appa} colors

    更新日期:2020-10-13
  • Datalog-Expressibility for Monadic and Guarded Second-Order Logic
    arXiv.cs.CC Pub Date : 2020-10-12
    Manuel Bodirsky; Simon Knäuer; Sebastian Rudolph

    We characterise the sentences in Monadic Second-order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class C of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all integers l,k, there exists a *canonical* Datalog program Pi of width (l,k), that is, a Datalog program

    更新日期:2020-10-13
  • The Complexity of the Homomorphism Problem for Boolean structures
    arXiv.cs.CC Pub Date : 2020-10-10
    Libor Barto; William DeMeo; Antoine Mottet

    We show that for a fixed Boolean structure $\mathscr B$ of arbitrary finite signature---i.e., not necessarily purely relational---the problem of deciding whether there exists a homomorphism to $\mathscr B$ is either in P or NP-complete.

    更新日期:2020-10-13
  • The Complexity of Mixed-Connectivity
    arXiv.cs.CC Pub Date : 2020-10-09
    Édouard Bonnet; Sergio Cabello

    We investigate the parameterized complexity in $a$ and $b$ of determining whether a graph~$G$ has a subset of $a$ vertices and $b$ edges whose removal disconnects $G$, or disconnects two prescribed vertices $s, t \in V(G)$.

    更新日期:2020-10-13
  • Symmetric Promise Constraint Satisfaction Problems: Beyond the Boolean Case
    arXiv.cs.CC Pub Date : 2020-10-09
    Libor Barto; Diego Battistelli; Kevin M. Berg

    The Promise Constraint Satisfaction Problem (PCSP) is a recently introduced vast generalization of the Constraint Satisfaction Problem (CSP). We investigate the computational complexity of a class of PCSPs beyond the most studied cases - approximation variants of satisfiability and graph coloring problems. We give an almost complete classification for the class of PCSPs of the form: given a 3-uniform

    更新日期:2020-10-12
  • Finitely Tractable Promise Constraint Satisfaction Problems
    arXiv.cs.CC Pub Date : 2020-10-09
    Kristina Asimi; Libor Barto

    The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS '19] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to

    更新日期:2020-10-12
  • Deterministic computation of the characteristic polynomial in the time of matrix multiplication
    arXiv.cs.CC Pub Date : 2020-10-09
    Vincent Neiger; Clément Pernet

    This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, to our knowledge, this was only achieved by resorting to genericity assumptions or randomization techniques, while the best known complexity bound with a general deterministic

    更新日期:2020-10-12
  • Constant-time connectivity tests
    arXiv.cs.CC Pub Date : 2020-10-09
    Philipp Klaus Krause

    We present implementations of constant-time algorithms for connectivity tests and related problems. Some are implementations of slightly improved variants of previously known algorithms; for other problems we present new algorithms that have substantially better runtime than previously known algorithms (estimates of the distance to and tolerant testers for connectivity, 2-edge-connectivity, 3-edge-connectivity

    更新日期:2020-10-12
  • Succinct Navigational Oracles for Families of Intersection Graphs on a Circle
    arXiv.cs.CC Pub Date : 2020-10-09
    Hüseyin Acan; Sankardeep Chakraborty; Seungbum Jo; Kei Nakashima; Kunihiko Sadakane; Srinivasa Rao Satti

    We consider the problem of designing succinct navigational oracles, i.e., succinct data structures supporting basic navigational queries such as degree, adjacency, and neighborhood efficiently for intersection graphs on a circle, which include graph classes such as {\it circle graphs}, {\it $k$-polygon-circle graphs}, {\it circle-trapezoid graphs}, {\it trapezoid graphs}. The degree query reports the

    更新日期:2020-10-12
  • Complexity of the list homomorphism problem in hereditary graph classes
    arXiv.cs.CC Pub Date : 2020-10-07
    Karolina Okrasa; Paweł Rzążewski

    A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. For a fixed graph $H$, in the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is equipped with a list $L(v) \subseteq V(H)$. We ask if there exists a homomorphism $f$ from $G$ to $H$, in which $f(v) \in L(v)$ for every $v \in V(G)$. Feder, Hell, and Huang

    更新日期:2020-10-08
  • Linear-time Temporal Logic with Team Semantics: Expressivity and Complexity
    arXiv.cs.CC Pub Date : 2020-10-07
    Jonni Virtema; Jana Hofmann; Bernd Finkbeiner; Juha Kontinen; Fan Yang

    We study the expressivity and the model checking problem of linear temporal logic with team semantics (TeamLTL). In contrast to LTL, TeamLTL is capable of defining hyperproperties, i.e., properties which relate multiple execution traces. Logics for hyperproperties have so far been mostly obtained by extending temporal logics like LTL and QPTL with trace quantification, resulting in HyperLTL and HyperQPTL

    更新日期:2020-10-08
  • Balanced incomplete block designs and exact satisfiability
    arXiv.cs.CC Pub Date : 2020-10-05
    Bernd. R. Schuh

    The paper explores the correspondence between balanced incomplete block designs (BIBD) and certain linear CNF formulas by identifying the points of a block design with the clauses of the Boolean formula and blocks with Boolean variables. Parallel classes in BIBDs correspond to XSAT solutions in the corresponding formula. This correspondence allows for transfers of results from one field to the other

    更新日期:2020-10-08
  • StoqMA vs. MA: the power of error reduction
    arXiv.cs.CC Pub Date : 2020-10-06
    Dorit Aharonov; Alex B. Grilo; Yupan Liu

    StoqMA characterizes the computational hardness of stoquastic local Hamiltonians, which is a family of Hamiltonians that not suffer from the sign problem. Although an error reduction is commonplace for many complexity classes, such as BPP, BQP, MA, QMA, etc., this property remains open for StoqMA since Bravyi, Bessen and Terhal defined this class in 2006. In this note, we show that an error reduction

    更新日期:2020-10-07
  • InstaHide: Instance-hiding Schemes for Private Distributed Learning
    arXiv.cs.CC Pub Date : 2020-10-06
    Yangsibo Huang; Zhao Song; Kai Li; Sanjeev Arora

    How can multiple distributed entities collaboratively train a shared deep net on their private data while preserving privacy? This paper introduces InstaHide, a simple encryption of training images, which can be plugged into existing distributed deep learning pipelines. The encryption is efficient and applying it during training has minor effect on test accuracy. InstaHide encrypts each training image

    更新日期:2020-10-07
  • Limitations of Sums of Bounded-Read Formulas
    arXiv.cs.CC Pub Date : 2020-10-03
    Purnata Ghosal; B. V. Raghavendra Rao

    Proving super polynomial size lower bounds for various classes of arithmetic circuits computing explicit polynomials is a very important and challenging task in algebraic complexity theory. We study representation of polynomials as sums of weaker models such as read once formulas (ROFs) and read once oblivious algebraic branching programs (ROABPs). We prove: (1) An exponential separation between sum

    更新日期:2020-10-06
  • Inapproximability for Local Correlation Clustering and Dissimilarity Hierarchical Clustering
    arXiv.cs.CC Pub Date : 2020-10-04
    Vaggos Chatziafratis; Neha Gupta; Euiwoong Lee

    We present hardness of approximation results for Correlation Clustering with local objectives and for Hierarchical Clustering with dissimilarity information. For the former, we study the local objective of Puleo and Milenkovic (ICML '16) that prioritizes reducing the disagreements at data points that are worst off and for the latter we study the maximization version of Dasgupta's cost function (STOC

    更新日期:2020-10-06
  • Burning Geometric Graphs
    arXiv.cs.CC Pub Date : 2020-10-03
    Arya Tanmay Gupta

    A procedure called \textit{graph burning} was introduced to facilitate the modelling of spread of an alarm, a social contagion, or a social influence or emotion on graphs and networks. Graph burning runs on discrete time-steps (or rounds). At each step $t$, first (a) an unburned vertex is burned (as a \textit{fire source}) from "outside", and then (b) the fire spreads to vertices adjacent to the vertices

    更新日期:2020-10-06
  • On the monophonic rank of graphs
    arXiv.cs.CC Pub Date : 2020-10-03
    Mitre C. Dourado; Vitor S. Ponciano; Rômulo L. O. da Silva

    A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em monophonic rank} of

    更新日期:2020-10-06
  • Tight Approximation Guarantees for Concave Coverage Problems
    arXiv.cs.CC Pub Date : 2020-10-02
    Siddharth Barman; Omar Fawzi; Paul Fermé

    In the maximum coverage problem, we are given subsets $T_1, \ldots, T_m$ of a universe $[n]$ along with an integer $k$ and the objective is to find a subset $S \subseteq [m]$ of size $k$ that maximizes $C(S) := \Big|\bigcup_{i \in S} T_i\Big|$. It is a classic result that the greedy algorithm for this problem achieves an optimal approximation ratio of $(1-e^{-1})$. In this work we consider a generalization

    更新日期:2020-10-05
  • The Complexity of Boolean State Separation (Technical Report)
    arXiv.cs.CC Pub Date : 2020-10-02
    Ronny Tredup; Evgeny Erofeev

    For a Boolean type of nets $\tau$, a transition system $A$ is synthesizeable into a $\tau$-net $N$ if and only if distinct states of $A$ correspond to distinct markings of $N$, and $N$ prevents a transition firing if there is no related transition in $A$. The former property is called $\tau$-state separation property ($\tau$-SSP) while the latter -- $\tau$-event/state separation property ($\tau$-ESSP)

    更新日期:2020-10-05
  • On Approximability of Clustering Problems Without Candidate Centers
    arXiv.cs.CC Pub Date : 2020-09-30
    Vincent Cohen-Addad; Karthik C. S.; Euiwoong Lee

    The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the

    更新日期:2020-10-02
  • Approximation of the Double Travelling Salesman Problem with Multiple Stacks
    arXiv.cs.CC Pub Date : 2020-09-30
    Laurent Alfandari; Sophie Toulouse

    The Double Travelling Salesman Problem with Multiple Stacks, DTSPMS, deals with the collect and delivery of n commodities in two distinct cities, where the pickup and the delivery tours are related by LIFO constraints. During the pickup tour, commodities are loaded into a container of k rows, or stacks, with capacity c. This paper focuses on computational aspects of the DTSPMS, which is NP-hard. We

    更新日期:2020-10-02
  • Monochromatic Triangles, Intermediate Matrix Products, and Convolutions
    arXiv.cs.CC Pub Date : 2020-09-30
    Andrea Lincoln; Adam Polak; Virginia Vassilevska Williams

    The most studied linear algebraic operation, matrix multiplication, has surprisingly fast $O(n^\omega)$ time algorithms for $\omega<2.373$. On the other hand, the $(\min,+)$ matrix product which is at the heart of many fundamental graph problems such as APSP, has received only minor improvements over its brute-force cubic running time and is widely conjectured to require $n^{3-o(1)}$ time. There is

    更新日期:2020-10-02
  • Tetris is NP-hard even with $O(1)$ rows or columns
    arXiv.cs.CC Pub Date : 2020-09-29
    Sualeh Asif; Michael Coulombe; Erik D. Demaine; Martin L. Demaine; Adam Hesterberg; Jayson Lynch; Mihir Singhal

    We prove that the classic falling-block video game Tetris (both survival and board clearing) remains NP-complete even when restricted to 8 columns, or to 4 rows, settling open problems posed over 15 years ago [BDH+04]. Our reduction is from 3-Partition, similar to the previous reduction for unrestricted board sizes, but with a better packing of buckets. On the positive side, we prove that 2-column

    更新日期:2020-10-02
  • Multidimensional Stable Roommates with Master List
    arXiv.cs.CC Pub Date : 2020-09-29
    Robert Bredereck; Klaus Heeger; Dušan Knop; Rolf Niedermeier

    Since the early days of research in algorithms and complexity, the computation of stable matchings is a core topic. While in the classic setting the goal is to match up two agents (either from different "gender" (this is Stable Marriage) or "unrestricted" (this is Stable Roommates)), Knuth [1976] triggered the study of three- or multidimensional cases. Here, we focus on the study of Multidimensional

    更新日期:2020-09-30
  • Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations
    arXiv.cs.CC Pub Date : 2020-09-29
    Christian Beck; Arnulf Jentzen; Thomas Kruse

    The approximative calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential equations (BSDEs), in the numerical approximation of solutions of semilinear parabolic partial differential equations (PDEs), in statistical physics, in optimal stopping

    更新日期:2020-09-30
  • Improved FPT Algorithms for Deletion to Forest-like Structures
    arXiv.cs.CC Pub Date : 2020-09-26
    Kishen N. Gowda; Aditya Lonkar; Fahad Panolan; Vraj Patel; Saket Saurabh

    The Feedback Vertex Set problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph $G$ and a non-negative integer $k$, the objective is to test whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a forest. After a long line of improvement, recently, Li and Nederlof [SODA, 2020] designed a randomized

    更新日期:2020-09-30
  • Reachability in Dynamical Systems with Rounding
    arXiv.cs.CC Pub Date : 2020-09-28
    Christel Baier; Florian Funke; Simon Jantsch; Toghrul Karimov; Engel Lefaucheux; Joël Ouaknine; Amaury Pouly; David Purser; Markus A. Whiteland

    We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix $M \in \mathbb{Q}^{d \times d}$, an initial vector $x\in\mathbb{Q}^{d}$, a granularity $g\in \mathbb{Q}_+$ and a rounding operation $[\cdot]$ projecting a vector of $\mathbb{Q}^{d}$ onto another vector whose every

    更新日期:2020-09-29
  • A note on weak near unanimity polymorphisms
    arXiv.cs.CC Pub Date : 2020-09-28
    Arash Rafiey

    We show that deciding whether a given relational structure $\mathcal{R}$ admits a weak near unanimity polymorphism is polynomial time solvable.

    更新日期:2020-09-29
  • The canonical directed tree decomposition and its applications to the directed disjoint paths problem
    arXiv.cs.CC Pub Date : 2020-09-28
    Archontia C. Giannopoulou; Ken-ichi Kawarabayashi; Stephan Kreutzer; O-joung Kwon

    The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the canonical tree decomposition theorem for digraphs. More precisely, we construct directed tree-decompositions of digraphs that distinguish all their tangles of order $k$

    更新日期:2020-09-29
  • Quantum soundness of the classical low individual degree test
    arXiv.cs.CC Pub Date : 2020-09-27
    Zhengfeng Ji; Anand Natarajan; Thomas Vidick; John Wright; Henry Yuen

    Low degree tests play an important role in classical complexity theory, serving as basic ingredients in foundational results such as $\mathsf{MIP} = \mathsf{NEXP}$ [BFL91] and the PCP theorem [AS98,ALM+98]. Over the last ten years, versions of these tests which are sound against quantum provers have found increasing applications to the study of nonlocal games and the complexity class~$\mathsf{MIP}^*$

    更新日期:2020-09-29
  • On the Approximability of Robust Network Design
    arXiv.cs.CC Pub Date : 2020-09-25
    Yacine Al-Najjar; Walid Ben-Ameur; Jeremie Leguay; Jocelyne Elias

    Considering the dynamic nature of traffic, the robust network design problem consists in computing the capacity to be reserved on each network link such that any demand vector belonging to a polyhedral set can be routed. The objective is either to minimize congestion or a linear cost. And routing freely depends on the demand. We first prove that the robust network design problem with minimum congestion

    更新日期:2020-09-28
  • Pebble-Depth
    arXiv.cs.CC Pub Date : 2020-09-24
    Liam Jordon; Philippe Moser

    In this paper we introduce a new formulation of Bennett's logical depth based on pebble transducers. This notion is defined based on the difference between the minimal length descriptional complexity of strings from the perspective of finite-state transducers and pebble transducers. Our notion of pebble-depth satisfies the three fundamental properties of depth: i.e. easy sequences and random sequences

    更新日期:2020-09-28
  • Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability
    arXiv.cs.CC Pub Date : 2020-09-25
    Tesshu Hanaka; Yasuaki Kobayashi; Yusuke Kobayashi; Tsuyoshi Yagita

    We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the problem remains NP-hard. Moreover, for co-bipartite graphs and for line graphs, the problem also remains NP-hard. On the positive side, we give an algorithm deciding

    更新日期:2020-09-28
  • Fine-grained complexity of the list homomorphism problem: feedback vertex set and cutwidth
    arXiv.cs.CC Pub Date : 2020-09-24
    Marta Piecyk; Paweł Rzążewski

    For graphs $G,H$, a homomorphism from $G$ to $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the list homomorphism problem, denoted by \textsc{LHom}($H$), we are given a graph $G$ and lists $L: V(G) \to 2^{V(H)}$, and we ask for a homomorphism from $G$ to $H$ which additionally respects the lists $L$. Very recently Okrasa, Piecyk, and Rz\k{a}\.zewski [ESA 2020] defined an invariant $i^*(H)$

    更新日期:2020-09-25
  • On Tractability of Ulams Metric in Highier Dimensions and Dually Related Hierarchies of Problems
    arXiv.cs.CC Pub Date : 2020-09-24
    Sebastian Bala; Andrzej Kozik

    The Ulam's metric is the minimal number of moves consisting in removal of one element from a permutation and its subsequent reinsertion in different place, to go between two given permutations. Thet elements that are not moved create longest common subsequence of permutations. Aldous and Diaconis, in their paper, pointed that Ulam's metric had been introduced in the context of questions concerning

    更新日期:2020-09-25
  • On One-way Functions and Kolmogorov Complexity
    arXiv.cs.CC Pub Date : 2020-09-24
    Yanyi Liu; Rafael Pass

    We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial $t(n)\geq (1+\varepsilon)n, \varepsilon>0$, the following are equivalent: - One-way functions exists (which in turn is equivalent to the existence of secure private-key encryption schemes, digital signatures, pseudorandom generators, pseudorandom functions, commitment schemes, and more); - $t$-time

    更新日期:2020-09-25
  • Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines
    arXiv.cs.CC Pub Date : 2020-09-24
    Martin Koutecký; Johannes Zink

    The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number $k$ of job types, but possibly the number of jobs

    更新日期:2020-09-25
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