Computational Complexity
New submissions
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New submissions for Wed, 15 May 24
- [1] arXiv:2405.08051 [pdf, ps, other]
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Title: P=NPAuthors: Zikang DengSubjects: Computational Complexity (cs.CC)
This paper investigates an extremely classic NP-complete problem: How to determine if a graph G, where each vertex has a degree of at most 4, can be 3-colorable(The research in this paper focuses on graphs G that satisfy the condition where the degree of each vertex does not exceed 4. To conserve space, it is assumed throughout the paper that graph G meets this condition by default.). The author has meticulously observed the relationship between the coloring problem and semidefinite programming, and has creatively constructed the corresponding semidefinite programming problem R(G) for a given graph G. The construction method of R(G) refers to Theorem 1.1 in the paper. I have obtained and proven the conclusion: A graph G is 3-colorable if and only if the objective function of its corresponding optimization problem R(G) is bounded, and when the objective function is bounded, its minimum value is 0.
- [2] arXiv:2405.08255 [pdf, ps, other]
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Title: Total Variation Distance for Product Distributions is $\#\mathsf{P}$-CompleteAuthors: Arnab Bhattacharyya, Sutanu Gayen, Kuldeep S. Meel, Dimitrios Myrisiotis, A. Pavan, N. V. VinodchandranComments: 5 pages. An extended version of this paper appeared in the proceedings of IJCAI 2023, under the title "On approximating total variation distance" (see this https URL and arXiv:2206.07209)Subjects: Computational Complexity (cs.CC)
We show that computing the total variation distance between two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as Kullback-Leibler, Chi-square, and Hellinger, which tensorize over the marginals leading to efficient algorithms.
- [3] arXiv:2405.08377 [pdf, other]
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Title: ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop PuzzlesAuthors: MIT Hardness Group, Josh Brunner, Della Hendrickson, Lily Chung, Erik D. Demaine, Andy TockmanComments: 34 pages, 41 figures. To appear at Fun with Algorithms 2024Subjects: Computational Complexity (cs.CC)
We prove that Hamiltonicity in maximum-degree-3 grid graphs (directed or undirected) is ASP-complete, i.e., it has a parsimonious reduction from every NP search problem (including a polynomial-time bijection between solutions). As a consequence, given k Hamiltonian cycles, it is NP-complete to find another; and counting Hamiltonian cycles is #P-complete. If we require the grid graph's vertices to form a full $m \times n$ rectangle, then we show that Hamiltonicity remains ASP-complete if the edges are directed or if we allow removing some edges (whereas including all undirected edges is known to be easy). These results enable us to develop a stronger "T-metacell" framework for proving ASP-completeness of rectangular puzzles, which requires building just a single gadget representing a degree-3 grid-graph vertex. We apply this general theory to prove ASP-completeness of 38 pencil-and-paper puzzles where the goal is to draw a loop subject to given constraints: Slalom, Onsen-meguri, Mejilink, Detour, Tapa-Like Loop, Kouchoku, Icelom; Masyu, Yajilin, Nagareru, Castle Wall, Moon or Sun, Country Road, Geradeweg, Maxi Loop, Mid-loop, Balance Loop, Simple Loop, Haisu, Reflect Link, Linesweeper; Vertex/Touch Slitherlink, Dotchi-Loop, Ovotovata, Building Walk, Rail Pool, Disorderly Loop, Ant Mill, Koburin, Mukkonn Enn, Rassi Silai, (Crossing) Ichimaga, Tapa, Canal View, Aqre, and Paintarea. The last 14 of these puzzles were not even known to be NP-hard. Along the way, we prove ASP-completeness of some simple forms of Tree-Residue Vertex-Breaking (TRVB), including planar multigraphs with degree-6 breakable vertices, or with degree-4 breakable and degree-1 unbreakable vertices.
- [4] arXiv:2405.08649 [pdf, other]
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Title: The computational power of discrete chemical reaction networks with bounded executionsSubjects: Computational Complexity (cs.CC)
Chemical reaction networks (CRNs) model systems where molecules interact according to a finite set of reactions such as $A + B \to C$, representing that if a molecule of $A$ and $B$ collide, they disappear and a molecule of $C$ is produced. CRNs can compute Boolean-valued predicates $\phi:\mathbb{N}^d \to \{0,1\}$ and integer-valued functions $f:\mathbb{N}^d \to \mathbb{N}$; for instance $X_1 + X_2 \to Y$ computes the function $\min(x_1,x_2)$.
We study the computational power of execution bounded CRNs, in which only a finite number of reactions can occur from the initial configuration (e.g., ruling out reversible reactions such as $A \rightleftharpoons B$). The power and composability of such CRNs depend crucially on some other modeling choices that do not affect the computational power of CRNs with unbounded executions, namely whether an initial leader is present, and whether (for predicates) all species are required to "vote" for the Boolean output. If the CRN starts with an initial leader, and can allow only the leader to vote, then all semilinear predicates and functions can be stably computed in $O(n \log n)$ parallel time by execution bounded CRNs.
However, if no initial leader is allowed, all species vote, and the CRN is "noncollapsing" (does not shrink from initially large to final $O(1)$ size configurations), then execution bounded CRNs are severely limited, able to compute only eventually constant predicates. A key tool is to characterize execution bounded CRNs as precisely those with a nonnegative linear potential function that is strictly decreased by every reaction, a result that may be of independent interest.
Cross-lists for Wed, 15 May 24
- [5] arXiv:2405.08584 (cross-list from cs.IT) [pdf, ps, other]
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Title: When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?Subjects: Information Theory (cs.IT); Computational Complexity (cs.CC)
The Gilbert--Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate $\epsilon^2$ has relative distance at least $\frac{1}{2} - O(\epsilon)$ with high probability. However, it is a major challenge to construct explicit codes with similar parameters.
One hope to derandomize the Gilbert--Varshamov construction is with code concatenation: We begin with a (hopefully explicit) outer code ${C}_\mathrm{out}$ over a large alphabet, and concatenate that with a small binary random linear code ${C}_\mathrm{in}$. It is known that when we use \emph{independent} small codes for each coordinate, then the result lies on the GV bound with high probability, but this still uses a lot of randomness. In this paper, we consider the question of whether code concatenation with a single random linear inner code ${C}_\mathrm{in}$ can lie on the GV bound; and if so what conditions on ${C}_\mathrm{out}$ are sufficient for this.
We show that first, there do exist linear outer codes ${C}_\mathrm{out}$ that are "good" for concatenation in this sense (in fact, most linear codes codes are good). We also provide two sufficient conditions for ${C}_\mathrm{out}$, so that if ${C}_\mathrm{out}$ satisfies these, ${C}_\mathrm{out}\circ {C}_\mathrm{in}$ will likely lie on the GV bound. We hope that these conditions may inspire future work towards constructing explicit codes ${C}_\mathrm{out}$. - [6] arXiv:2405.08787 (cross-list from cs.DS) [pdf, ps, other]
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Title: Explicit Orthogonal Arrays and Universal Hashing with Arbitrary ParametersSubjects: Data Structures and Algorithms (cs.DS); Computational Complexity (cs.CC); Combinatorics (math.CO); Statistics Theory (math.ST)
Orthogonal arrays are a type of combinatorial design that were developed in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing arrays of minimum size. Kuperberg, Lovett and Peled (2017) gave a non-constructive existence proof of orthogonal arrays whose size is near-optimal (i.e., within a polynomial of Rao's lower bound), leaving open the question of an algorithmic construction. We give the first explicit, deterministic, algorithmic construction of orthogonal arrays achieving near-optimal size for all parameters. Our construction uses algebraic geometry codes.
In pseudorandomness, the notions of $t$-independent generators or $t$-independent hash functions are equivalent to orthogonal arrays. Classical constructions of $t$-independent hash functions are known when the size of the codomain is a prime power, but very few constructions are known for an arbitrary codomain. Our construction yields algorithmically efficient $t$-independent hash functions for arbitrary domain and codomain.
Replacements for Wed, 15 May 24
- [7] arXiv:2202.01103 (replaced) [pdf, ps, other]
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Title: A New Temporal Interpretation of Cluster EditingComments: 27 pages, 2 figures. Extended abstract appeared at IWOCA 2022Subjects: Discrete Mathematics (cs.DM); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
- [8] arXiv:2405.00065 (replaced) [pdf, other]
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Title: From Linear to Linearizable Optimization: A Novel Framework with Applications to Stationary and Non-stationary DR-submodular OptimizationComments: arXiv admin note: text overlap with arXiv:2402.08621Subjects: Optimization and Control (math.OC); Computational Complexity (cs.CC); Machine Learning (cs.LG); Machine Learning (stat.ML)
- [9] arXiv:2405.07137 (replaced) [pdf, other]
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Title: Oracle Separation between Noisy Quantum Polynomial Time and the Polynomial HierarchySubjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC)
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