Breaking the PPSZ Barrier for Unique 3-SAT

The PPSZ algorithm by Paturi, Pudl\'ak, Saks, and Zane (FOCS 1998) is the fastest known algorithm for (Promise) Unique k-SAT. We give an improved algorithm with exponentially faster bounds for Unique 3-SAT. For uniquely satisfiable 3-CNF formulas, we do the following case distinction: We call a clause critical if exactly one literal is satisfied by the unique satisfying assignment. If a formula has many critical clauses, we observe that PPSZ by itself is already faster. If there are only few clauses allover, we use an algorithm by Wahlstr\"om (ESA 2005) that is faster than PPSZ in this case. Otherwise we have a formula with few critical and many non-critical clauses. Non-critical clauses have at least two literals satisfied; we show how to exploit this to improve PPSZ.Comment: 13 pages; major revision with simplified algorithm but slightly worse constant

Parameterized pre-coloring extension and list coloring problems

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V(G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λ_P: X → Q for X ⊆ V(G), decide whether λ_P can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O*(2^k)-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n-k colors for every v ∈ V(G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k²) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k²) colors

Application des harmoniques générées dans un jet de gaz : mesure de sections efficaces d´ionisation des états excités de l´hélium

Les sections efficaces d'ionisation des états excités de l'hélium 1s2p 1P et 1s3p 1P ont été mesurées au voisinage du seuil d'ionisation en observant la saturation de l'ionisation. Les états excités de l'hélium sont préparés par l'absorption résonnante d'une harmonique d'ordre élevée produite par un laser picoseconde accordable. Ces états sont ensuite ionisés par un faisceau sonde en absorbant un photon. La modification de la fréquence du faisceau sonde, du proche infrarouge à l'ultraviolet, nous a permis de déterminer la dépendance de la section efficace d'ionisation en fonction de l'énergie de l'électron arraché. Les résultats expérimentaux confirment quantitativement les travaux théoriques effectués auparavant

Hollow microspheres as targets for staged laser-driven proton acceleration

A coated hollow core microsphere is introduced as a novel target in ultra-intense laser-matter interaction experiments. In particular, it facilitates staged laser-driven proton acceleration by combining conventional target normal sheath acceleration (TNSA), power recycling of hot laterally spreading electrons and staging in a very simple and cheap target geometry. During TNSA of protons from one area of the sphere surface, laterally spreading hot electrons form a charge wave. Due to the spherical geometry, this wave refocuses on the opposite side of the sphere, where an opening has been laser micromachined. This leads to a strong transient charge separation field being set up there, which can post-accelerate those TNSA protons passing through the hole at the right time. Experimentally, the feasibility of using such targets is demonstrated. A redistribution is encountered in the experimental proton energy spectra, as predicted by particle-in-cell simulations and attributed to transient fields set up by oscillating currents on the sphere surface

Temporal coherence of ultrashort high-order harmonic pulses

We have studied the temporal coherence of high-order harmonics (up to the 15th order) produced by focusing 100 fs laser pulses into an argon gas jet. We measure the visibility of the interference fringes, produced when two spatially separated harmonic sources interfere in the far field, as a function of the time delay between the two sources. In general, we find long coherence times, comparable to the expected pulse durations of the harmonics. For some of the harmonics, the interference pattern exhibits two regions, with significantly different coherence times. These results are interpreted in terms of different electronic trajectories contributing to harmonic generation. © 1998 American Physical Society

Temporal coherence of high-order harmonics

Systematic studies of the temporal coherence properties of high-order harmonic radiation are presented. These complement our previous investigations [Bellini et al., Phys. Rev. Lett. 81, 297 (1998)], where we showed the separation of the far-field pattern of high-order harmonics into two distinct spatial regions with different coherence times. Here we show how the coherence time of the inner and outer regions changes as a function of the harmonic order, the laser intensity, and the focusing conditions. Good agreement with the predictions of the semiclassical model of harmonic generation is obtained. © 1999 The American Physical Society

Kaivannaisjätteen luokittelu pysyväksi – Louhinassa muodostuvat sivukivet

Kaivannaisjäte voidaan kaivannaisjäteasetuksen (379/2008) mukaisesti luokitella pysyväksi, kun kaivannaisjäte täyttää asetuksessa mainitut kriteerit. Oppaan tarkoituksena on auttaa kaivannaisalan toimijoita ja viranomaisia louhinnassa syntyvän kaivannaisjätteen luokittelussa pysyväksi. Oppaassa esitetään menettelyt kaivannaisteollisuudessa syntyvän sivukiven luokittelulle kaivannaisjäteasetuksen mukaisesti pysyväksi jokokansallisen pysyvien kivilajien luettelon tai tapauskohtaisen arvioinnin avulla

Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{\Omega(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(\Delta(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $\Delta(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $\chi(H)$ does not exceed $\operatorname{tw}(H)$ and $\Delta(H)+1$, it is natural to ask whether similar upper bounds with respect to $\chi(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(\chi(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page