Super Strong ETH Is True for PPSZ with Small Resolution Width

We construct k-CNFs with m variables on which the strong version of PPSZ k-SAT algorithm, which uses resolution of width bounded by O(√{log log m}), has success probability at most 2^{-(1-(1 + ε)2/k)m} for every ε > 0. Previously such a bound was known only for the weak PPSZ algorithm which exhaustively searches through small subformulas of the CNF to see if any of them forces the value of a given variable, and for strong PPSZ the best known previous upper bound was 2^{-(1-O(log(k)/k))m} (Pudlák et al., ICALP 2017)

Impatient PPSZ - A Faster Algorithm for CSP

PPSZ is the fastest known algorithm for (d,k)-CSP problems, for most values of d and k. It goes through the variables in random order and sets each variable randomly to one of the d colors, excluding those colors that can be ruled out by looking at few constraints at a time. We propose and analyze a modification of PPSZ: whenever all but 2 colors can be ruled out for some variable, immediately set that variable randomly to one of the remaining colors. We show that our new "impatient PPSZ" outperforms PPSZ exponentially for all k and all d ? 3 on formulas with a unique satisfying assignment

PPSZ is better than you think

PPSZ, for long time the fastest known algorithm for $k$-SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to $0$ or $1$, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being implied by a small set of clauses). We show that PPSZ performs exponentially better than previously known, for all $k \geq 3$. For Unique-$3$-SAT we bound its running time by $O(1.306973^{n})$, which is somewhat better than the algorithm of Hansen, Kaplan, Zamir, and Zwick, which runs in time $O(1.306995^n)$. Before that, the best known upper bound for Unique-$3$-SAT was $O(1.3070319^n)$. All improvements are achieved without changing the original PPSZ. The core idea is to pretend that PPSZ does not process the variables in uniformly random order, but according to a carefully designed distribution. We write "pretend" since this can be done without any actual change to the algorithm

Breaking the Log Barrier: a Novel Universal Restart Strategy for Faster Las Vegas Algorithms

Let $\mathcal{A}$ be a Las Vegas algorithm, i.e. an algorithm whose running time $T$ is a random variable drawn according to a certain probability distribution $p$. In 1993, Luby, Sinclair and Zuckerman [LSZ93] proved that a simple universal restart strategy can, for any probability distribution $p$, provide an algorithm executing $\mathcal{A}$ and whose expected running time is $O(\ell^\star_p\log\ell^\star_p)$, where $\ell^\star_p=\Theta\left(\inf_{q\in (0,1]}Q_p(q)/q\right)$ is the minimum expected running time achievable with full prior knowledge of the probability distribution $p$, and $Q_p(q)$ is the $q$-quantile of $p$. Moreover, the authors showed that the logarithmic term could not be removed for universal restart strategies and was, in a certain sense, optimal. In this work, we show that, quite surprisingly, the logarithmic term can be replaced by a smaller quantity, thus reducing the expected running time in practical settings of interest. More precisely, we propose a novel restart strategy that executes $\mathcal{A}$ and whose expected running time is $O\big(\inf_{q\in (0,1]}\frac{Q_p(q)}{q}\,\psi\big(\log Q_p(q),\,\log (1/q)\big)\big)$ where $\psi(a,b)=1+\min\left\{a+b,a\log^2 a,\,b\log^2 b\right\}$. This quantity is, up to a multiplicative factor, better than: 1) the universal restart strategy of [LSZ93], 2) any $q$-quantile of $p$ for $q\in(0,1]$, 3) the original algorithm, and 4) any quantity of the form $\phi^{-1}(\mathbb{E}[\phi(T)])$ for a large class of concave functions $\phi$. The latter extends the recent restart strategy of [Zam22] achieving $O\left(e^{\mathbb{E}[\ln(T)]}\right)$, and can be thought of as algorithmic reverse Jensen's inequalities. Finally, we show that the behavior of $\frac{t\phi''(t)}{\phi'(t)}$ at infinity controls the existence of reverse Jensen's inequalities by providing a necessary and a sufficient condition for these inequalities to hold.Comment: 13 pages, 0 figure

ワセキケイ ロンリシキ ノ ジュウソク カノウセイ モンダイ ニ タイスル アルゴリズム ノ カイリョウ

京都大学0048新制・課程博士博士(情報学)甲第12459号情博第213号新制||情||46(附属図書館)UT51-2006-J450京都大学大学院情報学研究科通信情報システム専攻(主査)教授 岩間 一雄, 教授 湯淺 太一, 教授 小野寺 秀俊学位規則第4条第1項該当Doctor of InformaticsKyoto UniversityDFA

Algorithmic Applications of Hypergraph and Partition Containers

We present a general method to convert algorithms into faster algorithms for almost-regular input instances. Informally, an almost-regular input is an input in which the maximum degree is larger than the average degree by at most a constant factor. This family of inputs vastly generalizes several families of inputs for which we commonly have improved algorithms, including bounded-degree inputs and random inputs. It also generalizes families of inputs for which we don't usually have faster algorithms, including regular-inputs of arbitrarily high degree and very dense inputs. We apply our method to achieve breakthroughs in exact algorithms for several central NP-Complete problems including $k$-SAT, Graph Coloring, and Maximum Independent Set. Our main tool is the first algorithmic application of the relatively new Hypergraph Container Method (Saxton and Thomason 2015, Balogh, Morris and Samotij 2015). This recent breakthrough, which generalizes an earlier version for graphs (Kleitman and Winston 1982, Sapozhenko 2001), has been used extensively in recent years in extremal combinatorics. An important component of our work is the generalization of (hyper-)graph containers to Partition Containers