We consider a problem of approximating the size of the largest clique in a graph, with a monotone circuit. Concretely, we focus on distinguishing a random Erdős-Renyi graph $\mathcal{G}_{n,p}$, with $p=n^{-\frac{2}{α-1}}$ chosen st. with high probability it does not even have an $α$-clique, from a random clique on $β$ vertices (where $α\leq β$). Using the approximation method of Razborov, Alon and Boppana showed in 1987 that as long as $\sqrtα β< n^{1-δ}/\log n$, this problem requires a monotone circuit of size $n^{Ω(δ\sqrtα)}$, implying a lower bound of $2^{\tildeΩ(n^{1/3})}$ for the exact version of the problem when $k\approx n^{2/3}$. Recently Cavalar, Kumar, and Rossman improved their result by showing the tight lower bound $n^{Ω(k)}$, in a limited range $k \leq n^{1/3}$, implying a comparable $2^{\tildeΩ(n^{1/3})}$ lower bound.
We combine the ideas of Cavalar, Kumar and Rossman with the recent breakthrough results on the sunflower conjecture by Alweiss, Lovett, Wu and Zhang to show that as long as $αβ< n^{1-δ}/\log n$, any monotone circuit rejecting $\mathcal{G}_{n,p}$ while accepting a $β$-clique needs to have size at least $n^{Ω(δ^2 α)}$; this implies a stronger $2^{\tildeΩ(\sqrt{n})}$ lower bound for the unrestricted version of the problem.
We complement this result with a construction of an explicit monotone circuit of size $O(n^{δ^2 α/2})$ which rejects $\mathcal{G}_{n,p}$, and accepts any graph containing $β$-clique whenever $β> n^{1-δ}$. Those two theorems explain the largest $β$-clique that can be distinguished from $\mathcal{G}_{n, 1/2}$: when $β> n / 2^{C \sqrt{\log n}}$, polynomial size circuit co do it, while for $β< n / 2^{ω(\sqrt{\log n})}$ every circuit needs size $n^{ω(1)}$.
Circuit lower bounds are important since it is believed that a super-polynomial circuit lower bound for a problem in NP implies that P!=NP. Razborov has proved superpolynomial lower bounds for monotone circuits by using method of approximation. By extending this approach, researchers have proved exponential lower bounds for the monotone network complexity of several different functions. But until now, no one could prove a non-linear lower bound for the non-monotone complexity of any Boolean function in NP. While we show that in this paper by replacement of each Not gates into constant 1 equiva…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Duke University（杜克大学）
We note the significance of hypergraphic planted clique (HPC) detection in the investigation of computational hardness for a range of tensor problems. We ask if more evidence for the computational hardness of HPC detection can be developed. In particular, we conjecture if it is possible to establish the equivalence of the computational hardness between HPC and PC detection.
University of Primorska（波莫瑞大学）；Kyushu University（九州大学）；Brandenburg University of Technology（勃兰登堡工业大学）；IAM, University of Primorska（波莫瑞大学应用数学研究所）；Osaka Metropolitan University（大阪公立大学）
This paper investigates the computational complexity of deciding whether the vertices of a graph can be partitioned into a disjoint union of cliques and a triangle-free subgraph. This problem is known to be $\NP$-complete on arbitrary graphs. We show that this problem remains $\NP$-complete even when restricted to planar graphs and perfect graphs.
In 1985, Razborov discovered a proof that the monotone circuit complexity of the clique problem is super-polynomial. Alon and Boppana improved the result into exponential lower bound exp(Ω(n / \log n)^{1/3})) of a monotone circuit C to compute cliques of size (1/4) (n / log n)^{2/3}, where n is the number of vertices in a graph. Both proofs are based on the method of approximations and Erdos and Rado's sunflower lemma. There has been an interest in further generalization of the proof scheme. In this paper, we present a new approach to show the exponential monotone complexity. Unlike the standa…
Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erdős-Rado sunflower conjecture. The recent breakthrough of Alweiss, Lovett, Wu and Zhang gives an improved bound on the maximum size of a $w$-set system that excludes a robust sunflower. In this paper, we use this result to obtain an $\exp(n^{1/2-o(1)})$ lower bound on the monotone circuit size of an explicit $n$-variate monotone function, improving the previous best known $\exp(n^{1/3-o(1)})$ due to And…
We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $μ\in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monot…
A function f : {0, 1}^n -> {0, 1} is said to be k-monotone if it flips between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. Our work is part of a renewed focus in understanding testability of properties characterized by freeness of arbitrary order patterns as a generalization of monotonicity. Recently, Canonne et al. (ITCS 2017) initiate the study of k-monotone functions in the area of property testing, and Newman et al. (SODA 2…
The recent emergence of fine-grained cryptography strongly motivates developing an average-case analogue of Fine-Grained Complexity (FGC). This paper defines new versions of OV, $k$SUM and zero-$k$-clique that are both worst-case and average-case fine-grained hard assuming the core hypotheses of FGC. We then use these as a basis for fine-grained hardness and average-case hardness of other problems. The new problems represent their inputs in a certain ``factored'' form. We call them ``factored''-OV, ``factored''-zero-$k$-clique and ``factored''-$3$SUM. We show that factored-$k$-OV and factored …
We study the computational complexity of two related problems: recovering a planted $q$-coloring in $G(n,1/2)$, and finding efficiently verifiable witnesses of non-$q$-colorability (a.k.a. refutations) in $G(n,1/2)$. Our main results show hardness for both these problems in a restricted-but-powerful class of algorithms based on computing low-degree polynomials in the inputs. The problem of recovering a planted $q$-coloring is equivalent to recovering $q$ disjoint planted cliques that cover all the vertices -- a potentially easier variant of the well-studied planted clique problem. Our first re…
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-ha…
We introduce a framework for proving lower bounds on computational problems over distributions against algorithms that can be implemented using access to a statistical query oracle. For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather than directly accessing samples. Most natural algorithms of interest in theory and in practice, e.g., moments-based methods, local search, standard iterative methods for convex optimization, MCMC and simulated annealing can …
The problem of finding longest common subsequence (LCS) is one of the fundamental problems in computer science, which finds application in fields such as computational biology, text processing, information retrieval, data compression etc. It is well known that (decision version of) the problem of finding the length of a LCS of an arbitrary number of input sequences (which we refer to as Multi-LCS problem) is NP-complete. Jiang and Li [SICOMP'95] showed that if Max-Clique is hard to approximate within a factor of $s$ then Multi-LCS is also hard to approximate within a factor of $Θ(s)$. By the N…
We consider the pattern detection problem in graphs: given a constant size pattern graph $H$ and a host graph $G$, determine whether $G$ contains a subgraph isomorphic to $H$. Our main results are: * We prove that if a pattern $H$ contains a $k$-clique subgraph, then detecting whether an $n$ node host graph contains a not necessarily induced copy of $H$ requires at least the time for detecting whether an $n$ node graph contains a $k$-clique. The previous result of this nature required that $H$ contains a $k$-clique which is disjoint from all other $k$-cliques of $H$. * We show that if the famo…
We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of CLIQUE: $s$-CLUB and $s$-CLIQUE, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and $γ$-COMPLETE SUBGRAPH in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that $s$-CLUB and…
We investigate the problem $\#\mathsf{IndSub}(Φ)$ of counting all induced subgraphs of size $k$ in a graph $G$ that satisfy a given property $Φ$. This continues the work of Jerrum and Meeks who proved the problem to be $\#\mathrm{W[1]}$-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ is hard for $\#\mathrm{W[…
We give a reduction from {\sc clique} to establish that sparse PCA is NP-hard. The reduction has a gap which we use to exclude an FPTAS for sparse PCA (unless P=NP). Under weaker complexity assumptions, we also exclude polynomial constant-factor approximation algorithms.
The class P is in fact a proper sub-class of NP. We explore topological properties of the Hamming space 2^[n] where [n]={1, 2,..., n}. With the developed theory, we show: (i) a theorem that is closely related to Erdos and Rado's sunflower lemma, and claims a stronger statement in most cases, (ii) a new approach to prove the exponential monotone circuit complexity of the clique problem, (iii) NC \ne NP through the impossibility of a Boolean circuit with poly-log depth to compute cliques, based on the construction of (ii), and (iv) P \ne NP through the exponential circuit complexity of the cliqu…
In the paper "On P versus NP," Lev Gordeev attempts to extend the method of approximation, which successfully proved exponential lower bounds for monotone circuits, to the case of De Morgan Normal (DMN) circuits. As in Razborov's proof of exponential lower bounds for monotone circuits, Gordeev's work is focused on the NP-complete problem CLIQUE. If successful in proving exponential DMN circuit lower bounds for CLIQUE, Gordeev would prove that P $\neq$ NP. However, we show that Gordeev makes a crucial mistake in Lemma 12. This mistake comes from only approximating operations over positive circu…
We review and critique Boyu Sima's paper, "A solution of the P versus NP problem based on specific property of clique function," (arXiv:1911.00722) which claims to prove that ${\rm P}\neq{\rm NP}$ by way of removing the gap between the nonmonotone circuit complexity and the monotone circuit complexity of the clique function. We first describe Sima's argument, and then we describe where and why it fails. Finally, we present a simple example that clearly demonstrates the failure.
In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing $W[1]$-hardness proofs for these problems, since XNLP-hardness implies $W[t]$-hardness for all $t$. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle …
The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of width k. It follows, from the definition of the polynomial, that the constant-width and the constant-depth hierarchies of monotone arithmetic circuits are infinite, both in the commutative and the noncommutative settings. 2. We prove hardness-randomness tradeoffs for identity testing constant-…
We show that, assuming the (deterministic) Exponential Time Hypothesis, distinguishing between a graph with an induced $k$-clique and a graph in which all k-subgraphs have density at most $1-ε$, requires $n^{\tilde Ω(log n)}$ time. Our result essentially matches the quasi-polynomial algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this problem, and is the first one to rule out an additive PTAS for Densest $k$-Subgraph. We further strengthen this result by showing that our lower bound continues to hold when, in the soundness case, even subgraphs smaller by a near-polynomial factor …
In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products. We say that an optimization problem $Π$ is direct product feasible if it is possible to efficiently aggregate any $k$ instances of $Π$ and form one large instance of $Π$ such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the $k$ smaller instances. Given a direct product feasible optimization problem $Π$, our hardness amplification theorem may be informally stated as follows: If there is a…
Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Wil…
We consider the weighted antimonotone and the weighted monotone satisfiability problems on normalized circuits of depth at most $t \geq 2$, abbreviated {\sc wsat$^-[t]$} and {\sc wsat$^+[t]$}, respectively. These problems model the weighted satisfiability of antimonotone and monotone propositional formulas (including weighted anitmonoone/monotone {\sc cnf-sat}) in a natural way, and serve as the canonical problems in the definition of the parameterized complexity hierarchy. We characterize the parameterized complexity of {\sc wsat$^-[t]$} and {\sc wsat$^+[t]$} with respect to the genus of the …
The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs
INSAIT, Sofia University “St. Kliment Ohridski”（索菲亚大学“圣克利门特·奥赫里德斯基”研究所）；Weizmann Institute of Science（魏茨曼科学研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges. The $t$-tessellability problem aims to decide whether there is a tessellation cover of the graph with $t$ tessellations. This problem is motivated by its applications to quantum walk models, in especial, the evolution operator of the staggered model is obtained from a graph tessellation cover. We establish upper bounds on the tessellation cover number given by the minimum between the chromatic index of the graph and the c…
We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices $n$, is significantly harder than the unweighted problem by presenting an $O(n^{3 - \varepsilon})$ lower bound on the size of the kernel, under the assumption that N…
Inference problems with conjectured statistical-computational gaps are ubiquitous throughout modern statistics, computer science and statistical physics. While there has been success evidencing these gaps from the failure of restricted classes of algorithms, progress towards a more traditional reduction-based approach to computational complexity in statistical inference has been limited. Existing reductions have largely been limited to inference problems with similar structure -- primarily mapping among problems representable as a sparse submatrix signal plus a noise matrix, which are similar …
The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomial-time algorithms. The OV-conjecture in moderate dimension states there is no $ε>0$ for which an $O(N^{2-ε})\mathrm{poly}(D)$ time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size $N$ that contains $D$-dimensional binary vectors. We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far…
Given query access to a monotone function $f\colon\{0,1\}^n\to\{0,1\}$ with certificate complexity $C(f)$ and an input $x^{\star}$, we design an algorithm that outputs a size-$C(f)$ subset of $x^{\star}$ certifying the value of $f(x^{\star})$. Our algorithm makes $O(C(f) \cdot \log n)$ queries to $f$, which matches the information-theoretic lower bound for this problem and resolves the concrete open question posed in the STOC '22 paper of Blanc, Koch, Lange, and Tan [BKLT22]. We extend this result to an algorithm that finds a size-$2C(f)$ certificate for a real-valued monotone function with $O…
The goal of the paper is to give fine-grained hardness results for the Subgraph Isomorphism (SI) problem for fixed size induced patterns $H$, based on the $k$-Clique hypothesis that the current best algorithms for Clique are optimal. Our first main result is that for any pattern graph $H$ that is a {\em core}, the SI problem for $H$ is at least as hard as $t$-Clique, where $t$ is the size of the largest clique minor of $H$. This improves (for cores) the previous known results [Dalirrooyfard-Vassilevska W. STOC'20] that the SI for $H$ is at least as hard as $k$-clique where $k$ is the size of t…
An RNA sequence is a string composed of four types of nucleotides, $A, C, G$, and $U$. The goal of the RNA folding problem is to find a maximum cardinality set of crossing-free pairs of the form $\{A,U\}$ or $\{C,G\}$ in a given RNA sequence. The problem is central in bioinformatics and has received much attention over the years. Abboud, Backurs, and Williams (FOCS 2015) demonstrated a conditional lower bound for a generalized version of the RNA folding problem based on a conjectured hardness of the $k$-clique problem. Their lower bound requires the RNA sequence to have at least 36 types of sy…
Given $n$ weighted points (positive or negative) in $d$ dimensions, what is the axis-aligned box which maximizes the total weight of the points it contains? The best known algorithm for this problem is based on a reduction to a related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in time $O(n^d)$. It was conjectured [Barbay et al., CCCG'13] that this runtime is tight up to subpolynomial factors. We answer this conjecture affirmatively by providing a matching conditional lower bound. We also provide conditional lower bounds for the special case when points are arranged in…
Tree-adjoining grammars are a generalization of context-free grammars that are well suited to model human languages and are thus popular in computational linguistics. In the tree-adjoining grammar recognition problem, given a grammar $Γ$ and a string $s$ of length $n$, the task is to decide whether $s$ can be obtained from $Γ$. Rajasekaran and Yooseph's parser (JCSS'98) solves this problem in time $O(n^{2ω})$, where $ω< 2.373$ is the matrix multiplication exponent. The best algorithms avoiding fast matrix multiplication take time $O(n^6)$. The first evidence for hardness was given by Satta (J.…
We study the planted clique problem in which a clique of size k is planted in an Erdős-Rényi graph G(n, 1/2), and one is interested in either detecting or recovering this planted clique. This problem is interesting because it is widely believed to show a statistical-computational gap at clique size k=sqrt{n}, and has emerged as the prototypical problem with such a gap from which average-case hardness of other statistical problems can be deduced. It also displays a tight computational connection between the detection and recovery variants, unlike other problems of a similar nature. This wide in…
The CFG recognition problem is: given a context-free grammar $\mathcal{G}$ and a string $w$ of length $n$, decide if $w$ can be obtained from $\mathcal{G}$. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in $O(n^ω)$ time, where $ω<2.373$ is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic $O(n^3/\log^3{n})$ complexity. Lee (JACM'01) provided evidence that fast matrix m…
University of Waterloo（滑铁卢大学）；Massachusetts Institute of Technology（麻省理工学院）；Columbia University（哥伦比亚大学）
Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. O…
