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Local Enumeration and Majority Lower Bounds arXiv.cs.CC Pub Date : 2024-03-14 Mohit Gurumukhani, Ramamohan Paturi, Michael Saks, Pavel Pudlák, Navid Talebanfard
Depth-3 circuit lower bounds and $k$-SAT algorithms are intimately related; the state-of-the-art $\Sigma^k_3$-circuit lower bound and the $k$-SAT algorithm are based on the same combinatorial theorem. In this paper we define a problem which reveals new interactions between the two. Define Enum($k$, $t$) problem as: given an $n$-variable $k$-CNF and an initial assignment $\alpha$, output all satisfying
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Spectral Lower Bounds for Local Search arXiv.cs.CC Pub Date : 2024-03-10 Simina Brânzei, Nicholas J. Recker
Local search is a powerful heuristic in optimization and computer science, the complexity of which has been studied in the white box and black box models. In the black box model, we are given a graph $G = (V,E)$ and oracle access to a function $f : V \to \mathbb{R}$. The local search problem is to find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as
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Simulating Weighted Automata over Sequences and Trees with Transformers arXiv.cs.CC Pub Date : 2024-03-12 Michael Rizvi, Maude Lizaire, Clara Lacroce, Guillaume Rabusseau
Transformers are ubiquitous models in the natural language processing (NLP) community and have shown impressive empirical successes in the past few years. However, little is understood about how they reason and the limits of their computational capabilities. These models do not process data sequentially, and yet outperform sequential neural models such as RNNs. Recent work has shown that these models
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Average-case deterministic query complexity of boolean functions with fixed weight arXiv.cs.CC Pub Date : 2024-03-06 Yuan Li, Haowei Wu, Yi Yang
We explore the $\textit{average-case deterministic query complexity}$ of boolean functions under the $\textit{uniform distribution}$, denoted by $\mathrm{D}_\mathrm{ave}(f)$, the minimum average depth of zero-error decision tree computing a boolean function $f$. This measure found several applications across diverse fields. We study $\mathrm{D}_\mathrm{ave}(f)$ of several common functions, including
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The complexity of computing in continuous time: space complexity is precision arXiv.cs.CC Pub Date : 2024-03-04 Manon Blanc, Olivier Bournez
Models of computations over the integers are equivalent from a computability and complexity theory point of view by the Church-Turing thesis. It is not possible to unify discrete-time models over the reals. The situation is unclear but simpler for continuous-time models, as there is a unifying mathematical model provided by ordinary differential equations (ODEs). For example, the GPAC model of Shannon
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Bounded Depth Frege Lower Bounds for Random 3-CNFs via Deterministic Restrictions arXiv.cs.CC Pub Date : 2024-03-04 Svyatoslav Gryaznov, Navid Talebanfard
A major open problem in proof complexity is to show that random 3-CNFs with linear number of clauses require super-polynomial size refutations in bounded depth Frege. We make a first step towards this question by showing a super-linear lower bound: for every $k$, there exists $\epsilon > 0$ such that any depth-$k$ Frege refutation of a random $n$-variable 3-CNF with $\Theta(n)$ clauses has $\Omega(n^{1
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Deterministic Weighted Automata under Partial Observability arXiv.cs.CC Pub Date : 2024-03-01 Jakub Michaliszyn, Jan Otop
Weighted automata is a basic tool for specification in quantitative verification, which allows to express quantitative features of analysed systems such as resource consumption. Quantitative specification can be assisted by automata learning as there are classic results on Angluin-style learning of weighted automata. The existing work assumes perfect information about the values returned by the target
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Tight Lower Bounds for Block-Structured Integer Programs arXiv.cs.CC Pub Date : 2024-02-27 Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Alexandra Lassota, Asaf Levin
We study fundamental block-structured integer programs called tree-fold and multi-stage IPs. Tree-fold IPs admit a constraint matrix with independent blocks linked together by few constraints in a recursive pattern; and transposing their constraint matrix yields multi-stage IPs. The state-of-the-art algorithms to solve these IPs have an exponential gap in their running times, making it natural to ask
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Aaronson-Ambainis Conjecture Is True For Random Restrictions arXiv.cs.CC Pub Date : 2024-02-21 Sreejata Kishor Bhattacharya
In an attempt to show that the acceptance probability of a quantum query algorithm making $q$ queries can be well-approximated almost everywhere by a classical decision tree of depth $\leq \text{poly}(q)$, Aaronson and Ambainis proposed the following conjecture: let $f: \{ \pm 1\}^n \rightarrow [0,1]$ be a degree $d$ polynomial with variance $\geq \epsilon$. Then, there exists a coordinate of $f$ with
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Quantum Automating $\mathbf{TC}^0$-Frege Is LWE-Hard arXiv.cs.CC Pub Date : 2024-02-15 Noel Arteche, Gaia Carenini, Matthew Gray
We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate $\mathbf{TC}^0$-Frege. This extends the line of results of Kraj\'i\v{c}ek and Pudl\'ak (Information and Computation, 1998), Bonet, Pitassi, and Ray (FOCS, 1997), and Bonet
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Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems under ETH arXiv.cs.CC Pub Date : 2024-02-15 Shuangle Li, Bingkai Lin, Yuwei Liu
In this paper we present a new gap-creating randomized self-reduction for parameterized Maximum Likelihood Decoding problem over $\mathbb{F}_p$ ($k$-MLD$_p$). The reduction takes a $k$-MLD$_p$ instance with $k\cdot n$ vectors as input, runs in time $f(k)n^{O(1)}$ for some computable function $f$, outputs a $(3/2-\varepsilon)$-Gap-$k'$-MLD$_p$ instance for any $\varepsilon>0$, where $k'=O(k^2\log k)$
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The Complexity of Promise Constraint Satisfaction Problem Seen from the Other Side arXiv.cs.CC Pub Date : 2024-02-09 Kristina Asimi, Libor Barto, Victor Dalmau
We introduce the framework of the left-hand side restricted promise constraint satisfaction problem, which includes problems like approximating clique number of a graph. We study the parameterized complexity of problems in this class and provide some initial results. The main technical contribution is a sufficient condition for W[1]-hardness which, in particular, covers left-hand side restricted bounded
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Nearest Neighbor Complexity and Boolean Circuits arXiv.cs.CC Pub Date : 2024-02-09 Mason DiCicco, Vladimir Podolskii, Daniel Reichman
A nearest neighbor representation of a Boolean function $f$ is a set of vectors (anchors) labeled by $0$ or $1$ such that $f(\vec{x}) = 1$ if and only if the closest anchor to $x$ is labeled by $1$. This model was introduced by Hajnal, Liu, and Tur\'an (2022), who studied bounds on the number of anchors required to represent Boolean functions under different choices of anchors (real vs. Boolean vectors)
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Simple inexpensive vertex and edge invariants distinguishing dataset strongly regular graphs arXiv.cs.CC Pub Date : 2024-02-07 Jarek Duda
While standard Weisfeiler-Leman vertex labels are not able to distinguish even vertices of regular graphs, there is proposed and tested family of inexpensive polynomial time vertex and edge invariants, distinguishing much more difficult SRGs (strongly regular graphs), also often their vertices. Among 43717 SRGs from dataset by Edward Spence, proposed vertex invariants alone were able to distinguish
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Exponential Separation Between Powers of Regular and General Resolution Over Parities arXiv.cs.CC Pub Date : 2024-02-06 Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, Pavel Dvořák
Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities, is an outstanding problem that has received a lot of attention after its introduction by Raz and Tzamaret [Ann. Pure Appl. Log.'08]. Very recently, Efremenko, Garl\'ik and Itsykson [ECCC'23] proved the first exponential lower bounds on the size of ResLin proofs that were
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XNLP-hardness of Parameterized Problems on Planar Graphs arXiv.cs.CC Pub Date : 2024-02-05 Hans L. Bodlaender, Krisztina Szilágyi
The class XNLP consists of (parameterized) problems that can be solved nondeterministically in $f(k)n^{O(1)}$ time and $f(k)\log n$ space, where $n$ is the size of the input instance and $k$ the parameter. The class XALP consists of problems that can be solved in the above time and space with access to an additional stack. These two classes are a "natural home" for many standard graph problems and
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Ruling Out Low-rank Matrix Multiplication Tensor Decompositions with Symmetries via SAT arXiv.cs.CC Pub Date : 2024-02-01 Jason Yang
We analyze rank decompositions of the $3\times 3$ matrix multiplication tensor over $\mathbb{Z}/2\mathbb{Z}$. We restrict our attention to decompositions of rank $\le 21$, as only those decompositions will yield an asymptotically faster algorithm for matrix multiplication than Strassen's algorithm. To reduce search space, we also require decompositions to have certain symmetries. Using Boolean SAT
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Hausdorff Reductions and the Exponential Hierarchies arXiv.cs.CC Pub Date : 2024-02-01 Enrico MaliziaUniversity of Bologna, Italy
The Strong Exponential Hierarchy $SEH$ was shown to collapse to $P^{NExp}$ by Hemachandra by proving $P^{NExp} = NP^{NExp}$ via a census argument. Nonetheless, Hemachandra also asked for certificate-based and alternating Turing machine characterizations of the $SEH$ levels, in the hope that these might have revealed deeper structural reasons behind the collapse. These open questions have thus far remained
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Hardness of Random Reordered Encodings of Parity for Resolution and CDCL arXiv.cs.CC Pub Date : 2024-02-01 Leroy Chew, Alexis de Colnet, Friedrich Slivovsky, Stefan Szeider
Parity reasoning is challenging for Conflict-Driven Clause Learning (CDCL) SAT solvers. This has been observed even for simple formulas encoding two contradictory parity constraints with different variable orders (Chew and Heule 2020). We provide an analytical explanation for their hardness by showing that they require exponential resolution refutations with high probability when the variable order
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On Small-depth Frege Proofs for PHP arXiv.cs.CC Pub Date : 2024-01-28 Johan Håstad
We study Frege proofs for the one-to-one graph Pigeon Hole Principle defined on the $n\times n$ grid where $n$ is odd. We are interested in the case where each formula in the proof is a depth $d$ formula in the basis given by $\land$, $\lor$, and $\neg$. We prove that in this situation the proof needs to be of size exponential in $n^{\Omega (1/d)}$. If we restrict the size of each line in the proof
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Structure in Communication Complexity and Constant-Cost Complexity Classes arXiv.cs.CC Pub Date : 2024-01-26 Hamed Hatami, Pooya Hatami
Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often imply a bound on some matrix parameter. The challenge
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The Boundaries of Tractability in Hierarchical Task Network Planning arXiv.cs.CC Pub Date : 2024-01-25 Cornelius Brand, Robert Ganian, Fionn Mc Inerney, Simon Wietheger
We study the complexity-theoretic boundaries of tractability for three classical problems in the context of Hierarchical Task Network Planning: the validation of a provided plan, whether an executable plan exists, and whether a given state can be reached by some plan. We show that all three problems can be solved in polynomial time on primitive task networks of constant partial order width (and a generalization
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On Pigeonhole Principles and Ramsey in TFNP arXiv.cs.CC Pub Date : 2024-01-23 Siddhartha Jain, Jiawei Li, Robert Robere, Zhiyang Xun
The generalized pigeonhole principle says that if tN + 1 pigeons are put into N holes then there must be a hole containing at least t + 1 pigeons. Let t-PPP denote the class of all total NP-search problems reducible to finding such a t-collision of pigeons. We introduce a new hierarchy of classes defined by the problems t-PPP. In addition to being natural problems in TFNP, we show that classes in and
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CLIQUE as an AND of Polynomial-Sized Monotone Constant-Depth Circuits arXiv.cs.CC Pub Date : 2024-01-22 Levente Bodnár
This paper shows that calculating $k$-CLIQUE on $n$ vertex graphs, requires the AND of at least $2^{n/4k}$ monotone, constant-depth, and polynomial-sized circuits, for sufficiently large values of $k$. The proof relies on a new, monotone, one-sided switching lemma, designed for cliques.
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What Juris Hartmanis taught me about Reductions arXiv.cs.CC Pub Date : 2024-01-20 Neil Immerman
I was a student of Juris Hartmanis at Cornell in the late 1970's. He believed that there was great potential in studying restricted reductions. I describe here some of his influences on me and, in particular, how his insights concerning reductions helped me to prove that nondeterministic space is closed under complementation.
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Unambiguous parity-query complexity arXiv.cs.CC Pub Date : 2024-01-20 Dmytro Gavinsky
We give a lower bound of $\Omega(\sqrt n)$ on the unambiguous randomised parity-query complexity of the approximate majority problem -- that is, on the lowest randomised parity-query complexity of any function over $\{0,1\}^n$ whose value is "0" if the Hamming weight of the input is at most n/3, is "1" if the weight is at least 2n/3, and may be arbitrary otherwise.
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The Bit Complexity of Dynamic Algebraic Formulas and their Determinants arXiv.cs.CC Pub Date : 2024-01-20 Emile Anand, Jan van den Brand, Mehrdad Ghadiri, Daniel Zhang
Many iterative algorithms in optimization, computational geometry, computer algebra, and other areas of computer science require repeated computation of some algebraic expression whose input changes slightly from one iteration to the next. Although efficient data structures have been proposed for maintaining the solution of such algebraic expressions under low-rank updates, most of these results are
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Isomorphism Testing of Rooted Trees in Linear Time arXiv.cs.CC Pub Date : 2024-01-15 Anna Lindeberg
The AHU-algorithm solves the computationally difficult graph isomorphism problem for rooted trees, and does so with a linear time complexity. Although the AHU-algorithm has remained state of the art for almost 50 years, it has been criticized for being unclearly presented, and no complete proof of correctness has been given. In this text, that gap is filled: we formalize the algorithm's main point
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Low-Rank Tensor Decomposition over Finite Fields arXiv.cs.CC Pub Date : 2024-01-12 Jason Yang
We show that finding rank-1, rank-2, and rank-3 decompositions of a 3D tensor over a fixed finite field can be done in polynomial time. However, if some cells in the tensor are allowed to have arbitrary values, then rank-2 is NP-hard over the integers modulo 2. We also explore rank-1 decomposition of a 3D tensor and of a matrix where some cells are allowed to have arbitrary values.
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On the Exact Matching Problem in Dense Graphs arXiv.cs.CC Pub Date : 2024-01-08 Nicolas El Maalouly, Sebastian Haslebacher, Lasse Wulf
In the Exact Matching problem, we are given a graph whose edges are colored red or blue and the task is to decide for a given integer k, if there is a perfect matching with exactly k red edges. Since 1987 it is known that the Exact Matching Problem can be solved in randomized polynomial time. Despite numerous efforts, it is still not known today whether a deterministic polynomial-time algorithm exists
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Lower Bounds on Cardinality of Reducts for Decision Tables from Closed Classes arXiv.cs.CC Pub Date : 2024-01-02 Azimkhon Ostonov, Mikhail Moshkov
In this paper, we consider classes of decision tables closed under removal of attributes (columns) and changing of decisions attached to rows. For decision tables from closed classes, we study lower bounds on the minimum cardinality of reducts, which are minimal sets of attributes that allow us to recognize, for a given row, the decision attached to it. We assume that the number of rows in decision
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Probabilistically Checkable Reconfiguration Proofs and Inapproximability of Reconfiguration Problems arXiv.cs.CC Pub Date : 2023-12-31 Shuichi Hirahara, Naoto Ohsaka
Motivated by the inapproximability of reconfiguration problems, we present a new PCP-type characterization of PSPACE, which we call a probabilistically checkable reconfiguration proof (PCRP): Any PSPACE computation can be encoded into an exponentially long sequence of polynomially long proofs such that every adjacent pair of the proofs differs in at most one bit, and every proof can be probabilistically
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Complexity-Theoretic Implications of Multicalibration arXiv.cs.CC Pub Date : 2023-12-28 Sílvia Casacuberta, Cynthia Dwork, Salil Vadhan
We present connections between the recent literature on multigroup fairness for prediction algorithms and classical results in computational complexity. Multiaccurate predictors are correct in expectation on each member of an arbitrary collection of pre-specified sets. Multicalibrated predictors satisfy a stronger condition: they are calibrated on each set in the collection. Multiaccuracy is equivalent
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Rethinking Model-based, Policy-based, and Value-based Reinforcement Learning via the Lens of Representation Complexity arXiv.cs.CC Pub Date : 2023-12-28 Guhao Feng, Han Zhong
Reinforcement Learning (RL) encompasses diverse paradigms, including model-based RL, policy-based RL, and value-based RL, each tailored to approximate the model, optimal policy, and optimal value function, respectively. This work investigates the potential hierarchy of representation complexity -- the complexity of functions to be represented -- among these RL paradigms. We first demonstrate that,
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On Inapproximability of Reconfiguration Problems: PSPACE-Hardness and some Tight NP-Hardness Results arXiv.cs.CC Pub Date : 2023-12-28 Karthik C. S., Pasin Manurangsi
The field of combinatorial reconfiguration studies search problems with a focus on transforming one feasible solution into another. Recently, Ohsaka [STACS'23] put forth the Reconfiguration Inapproximability Hypothesis (RIH), which roughly asserts that there is some $\varepsilon>0$ such that given as input a $k$-CSP instance (for some constant $k$) over some constant sized alphabet, and two satisfying
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Approximation algorithms for noncommutative constraint satisfaction problems arXiv.cs.CC Pub Date : 2023-12-28 Eric Culf, Hamoon Mousavi, Taro Spirig
We study operator - or noncommutative - variants of constraint satisfaction problems (CSPs). These higher-dimensional variants are a core topic of investigation in quantum information, where they arise as nonlocal games and entangled multiprover interactive proof systems (MIP*). The idea of higher-dimensional relaxations of CSPs is also important in the classical literature. For example since the celebrated
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Computing Balanced Solutions for Large International Kidney Exchange Schemes When Cycle Length Is Unbounded arXiv.cs.CC Pub Date : 2023-12-27 Márton Benedek, Péter Biró, Gergely Csáji, Matthew Johnson, Daniël Paulusma, Xin Ye
In kidney exchange programmes (KEP) patients may swap their incompatible donors leading to cycles of kidney transplants. Nowadays, countries try to merge their national patient-donor pools leading to international KEPs (IKEPs). As shown in the literature, long-term stability of an IKEP can be achieved through a credit-based system. In each round, every country is prescribed a "fair" initial allocation
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The linear time encoding scheme fails to encode arXiv.cs.CC Pub Date : 2023-12-26 Yotam Dikstein, Irit Dinur, Shiri Sivan
We point out an error in the paper "Linear Time Encoding of LDPC Codes" (by Jin Lu and Jos\'e M. F. Moura, IEEE Trans). The paper claims to present a linear time encoding algorithm for every LDPC code. We present a family of counterexamples, and point out where the analysis fails. The algorithm in the aforementioned paper fails to encode our counterexample, let alone in linear time.
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Exponential Lower Bounds for Sums of ROABPs arXiv.cs.CC Pub Date : 2023-12-26 Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, Amir Shpilka
In this paper, we prove the first \emph{super-polynomial} and, in fact, \emph{exponential} lower bound for the model of \emph{sum of read-once oblivious algebraic branching programs} (ROABPs). In particular, we give an explicit polynomial such that any sum of ROABPs (equivalently, sum of \emph{ordered} set-multilinear branching programs, each with a possibly different ordering) computing it must have
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The Zeta ($ζ$) Notation for Complex Asymptotes arXiv.cs.CC Pub Date : 2023-12-24 Anurag Dutta, K. Lakshmanan, John Harshith, A. Ramamoorthy
Time Complexity is an important metric to compare algorithms based on their cardinality. The commonly used, trivial notations to qualify the same are the Big-Oh, Big-Omega, Big-Theta, Small-Oh, and Small-Omega Notations. All of them, consider time a part of the real entity, i.e., Time coincides with the horizontal axis in the argand plane. But what if the Time rather than completely coinciding with
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On the Existence of Seedless Condensers: Exploring the Terrain arXiv.cs.CC Pub Date : 2023-12-22 Eshan Chattopadhyay, Mohit Gurumukhani, Noam Ringach
We prove several new results for seedless condensers in the context of three related classes of sources: NOSF sources, SHELA sources as defined by [AORSV, EUROCRYPT'20], and almost CG sources as defined by [DMOZ, STOC'23]. We will think of these sources as a sequence of random variables $\mathbf{X}=\mathbf{X}_1,\dots,\mathbf{X}_\ell$ on $\ell$ symbols where at least $g$ symbols are "good" (i.e., uniformly
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The Problem of Computational Complexity arXiv.cs.CC Pub Date : 2023-12-20 Rami Zaidan
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century. Second, building on the theory of functional relations in mathematics, it provides a theoretical framework where we can rigorously distinguish two pairs of concepts:
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Leakage-Resilient Hardness Equivalence to Logspace Derandomization arXiv.cs.CC Pub Date : 2023-12-21 Yakov Shalunov
Efficient derandomization has long been a goal in complexity theory, and a major recent result by Yanyi Liu and Rafael Pass identifies a new class of hardness assumption under which it is possible to perform time-bounded derandomization efficiently: that of ''leakage-resilient hardness.'' They identify a specific form of this assumption which is $\textit{equivalent}$ to $\mathsf{prP} = \mathsf{prBPP}$
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Enumerating Defensive Alliances arXiv.cs.CC Pub Date : 2023-12-19 Zhidan Feng, Henning Fernau, Kevin Mann
In this paper, we study the task of enumerating (and counting) locally and globally minimal defensive alliances in graphs. We consider general graphs as well as special graph classes. From an input-sensitive perspective, our presented algorithms are mostly optimal.
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Offensive Alliances in Signed Graphs arXiv.cs.CC Pub Date : 2023-12-19 Zhidan Feng, Henning Fernau, Kevin Mann, Xingqin Qi
Signed graphs have been introduced to enrich graph structures expressing relationships between persons or general social entities, introducing edge signs to reflect the nature of the relationship, e.g., friendship or enmity. Independently, offensive alliances have been defined and studied for undirected, unsigned graphs. We join both lines of research and define offensive alliances in signed graphs
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Almost Uniform Sampling of Independent Sets in Polynomial Time -- Implying NP=RP arXiv.cs.CC Pub Date : 2023-12-19 Andras Farago
We prove the unexpected result that almost uniform sampling of independent sets in graphs is possible via a probabilistic polynomial time algorithm. Note that our sampling algorithm (if correct) has extremely surprising consequences; the most important one being no less than the unlikely collapse NP=RP.
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FPT Approximation using Treewidth: Capacitated Vertex Cover, Target Set Selection and Vector Dominating Set arXiv.cs.CC Pub Date : 2023-12-19 Bingkai Lin, Huairui Chu
Treewidth is a useful tool in designing graph algorithms. Although many NP-hard graph problems can be solved in linear time when the input graphs have small treewidth, there are problems which remain hard on graphs of bounded treewidth. In this paper, we consider three vertex selection problems that are W[1]-hard when parameterized by the treewidth of the input graph, namely the capacitated vertex
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Unravelling Expressive Delegations: Complexity and Normative Analysis arXiv.cs.CC Pub Date : 2023-12-19 Giannis Tyrovolas, Andrei Constantinescu, Edith Elkind
We consider binary group decision-making under a rich model of liquid democracy recently proposed by Colley, Grandi, and Novaro (2022): agents submit ranked delegation options, where each option may be a function of multiple agents' votes; e.g., "I vote yes if a majority of my friends vote yes." Such ballots are unravelled into a profile of direct votes by selecting one entry from each ballot so as
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Refuting approaches to the log-rank conjecture for XOR functions arXiv.cs.CC Pub Date : 2023-12-14 Hamed Hatami, Kaave Hosseini, Shachar Lovett, Anthony Ostuni
The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function
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Approximation Algorithms for Preference Aggregation Using CP-Nets arXiv.cs.CC Pub Date : 2023-12-14 Abu Mohammmad Hammad Ali, Boting Yang, Sandra Zilles
This paper studies the design and analysis of approximation algorithms for aggregating preferences over combinatorial domains, represented using Conditional Preference Networks (CP-nets). Its focus is on aggregating preferences over so-called \emph{swaps}, for which optimal solutions in general are already known to be of exponential size. We first analyze a trivial 2-approximation algorithm that simply
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Algorithms and Complexity for Congested Assignments arXiv.cs.CC Pub Date : 2023-12-12 Jiehua Chen, Jiong Guo, Yinghui Wen
We study the congested assignment problem as introduced by Bogomolnaia and Moulin (2023). We show that deciding whether a competitive assignment exists can be done in polynomial time, while deciding whether an envy-free assignment exists is NP-complete.
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On Computability of Computable Problems arXiv.cs.CC Pub Date : 2023-12-13 Asad Khaliq
Computational problems are classified into computable and uncomputable problems.If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable.Turing machines can execute any algorithm therefore every computable problem is Turing computable.There are some variants of Turing machine that appear computationally more powerful but all
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Parallel Repetition of k-Player Projection Games arXiv.cs.CC Pub Date : 2023-12-08 Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, Dor Minzer
We study parallel repetition of k-player games where the constraints satisfy the projection property. We prove exponential decay in the value of a parallel repetition of projection games with value less than 1.
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Towards Tight Bounds for the Graph Homomorphism Problem Parameterized by Cutwidth via Asymptotic Rank Parameters arXiv.cs.CC Pub Date : 2023-12-06 Carla Groenland, Isja Mannens, Jesper Nederlof, Marta Piecyk, Paweł Rzążewski
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the graph homomorphism problem, denoted by $Hom(H)$, the graph $H$ is fixed and we need to determine if there exists a homomorphism from an instance graph $G$ to $H$. We study the complexity of the problem parameterized by the cutwidth of $G$. We aim, for each $H$, for algorithms for $Hom(H)$ running
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PLS is contained in PLC arXiv.cs.CC Pub Date : 2023-12-07 Takashi Ishizuka
Recently, Pasarkar, Papadimitriou, and Yannakakis (ITCS 2023) have introduced the new TFNP subclass called PLC that contains the class PPP; they also have proven that several search problems related to extremal combinatorial principles (e.g., Ramsey's theorem and the Sunflower lemma) belong to PLC. This short paper shows that the class PLC also contains PLS, a complexity class for TFNP problems that
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On Czerwinski's "${\rm P} \neq {\rm NP}$ relative to a ${\rm P}$-complete oracle" arXiv.cs.CC Pub Date : 2023-12-07 Michael C. Chavrimootoo, Tran Duy Anh Le, Michael P. Reidy, Eliot J. Smith
In this paper, we take a closer look at Czerwinski's "${\rm P}\neq{\rm NP}$ relative to a ${\rm P}$-complete oracle" [Cze23]. There are (uncountably) infinitely-many relativized worlds where ${\rm P}$ and ${\rm NP}$ differ, and it is well-known that for any ${\rm P}$-complete problem $A$, ${\rm P}^A \neq {\rm NP}^A \iff {\rm P}\neq {\rm NP}$. The paper defines two sets ${\rm D}_{\rm P}$ and ${\rm D}_{\rm
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When Input Integers are Given in the Unary Numeral Representation arXiv.cs.CC Pub Date : 2023-12-07 Tomoyuki Yamakami
Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a basis unit to measure the computational complexity of the problems. In sharp contrast, the "unarization" (or the "unary" numeral representation) of numbers has
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Enumerating Complexity Revisited arXiv.cs.CC Pub Date : 2023-12-07 Alexander Shekhovtsov, Georgii Zakharov
We reduce the best-known upper bound on the length of a program that enumerates a set in terms of the probability of it being enumerated by a random program. We prove a general result that any linear upper bound for finite sets implies the same linear bound for infinite sets. So far, the best-known upper bound was given by Solovay. He showed that the minimum length of a program enumerating a subset
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Canonization of a random graph by two matrix-vector multiplications arXiv.cs.CC Pub Date : 2023-12-06 Oleg Verbitsky, Maksim Zhukovskii
We show that a canonical labeling of a random $n$-vertex graph can be obtained by assigning to each vertex $x$ the triple $(w_1(x),w_2(x),w_3(x))$, where $w_k(x)$ is the number of walks of length $k$ starting from $x$. This takes time $O(n^2)$, where $n^2$ is the input size, by using just two matrix-vector multiplications. The linear-time canonization of a random graph is the classical result of Babai
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XOR Lemmas for Communication via Marginal Information arXiv.cs.CC Pub Date : 2023-12-05 Siddharth Iyer, Anup Rao
We define the $\textit{marginal information}$ of a communication protocol, and use it to prove XOR lemmas for communication complexity. We show that if every $C$-bit protocol has bounded advantage for computing a Boolean function $f$, then every $\tilde \Omega(C \sqrt{n})$-bit protocol has advantage $\exp(-\Omega(n))$ for computing the $n$-fold xor $f^{\oplus n}$. We prove exponentially small bounds