scholarly journals Parameterized complexity of synchronization and road coloring

2015 ◽  
Vol Vol. 17 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Vojtěch Vorel ◽  
Adam Roman
Keyword(s):  
Parameterized Complexity ◽  
Maximum Length ◽  
Parameter Analysis ◽  
Polynomial Kernel ◽  
Polynomial Hierarchy ◽  
Alphabet Size ◽  
International Audience

Automata, Logic and Semantics International audience First, we close the multi-parameter analysis of a canonical problem concerning short reset words (SYN) initiated by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multi-parameter analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum length of a reset word.

scholarly journals Parameterized Complexity of Directed Spanner Problems

Algorithmica ◽  
2021 ◽  
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
William Lochet ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Directed Graph ◽  
Parameterized Complexity ◽  
Directed Graphs ◽  
Directed Acyclic Graphs ◽  
Polynomial Kernel ◽  
Distortion Parameter ◽  
Spanning Subgraph ◽  
Acyclic Graphs ◽  
Positive Integers ◽  
Additive Distortion

AbstractWe initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k.

scholarly journals The Parameterized Complexity of Motion Planning for Snake-Like Robots

2019 ◽  
Author(s):  
Siddharth Gupta ◽  
Guy Sa'ar ◽  
Meirav Zehavi
Keyword(s):  
Motion Planning ◽  
Parameterized Complexity ◽  
Initial Position ◽  
Polynomial Kernel ◽  
Planning Problem ◽  
Color Coding ◽  
Grid Graphs ◽  
Input Size ◽  
Motion Problems ◽  
General Graphs

We study a motion-planning problem inspired by the game Snake that models scenarios like the transportation of linked wagons towed by a locomotor to the movement of a group of agents that travel in an ``ant-like'' fashion. Given a ``snake-like'' robot with initial and final positions in an environment modeled by a graph, our goal is to decide whether the robot can reach the final position from the initial position without intersecting itself. Already on grid graphs, this problem is PSPACE-complete [Biasi and Ophelders, 2018]. Nevertheless, we prove that even on general graphs, it is solvable in time k^{O(k)}|I|^{O(1)} where k is the size of the robot, and |I| is the input size. Towards this, we give a novel application of color-coding to sparsify the configuration graph of the problem. We also show that the problem is unlikely to have a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion problems has been~largely~neglected, thus our work is pioneering in this regard.

scholarly journals On the Structure of Valiant's Complexity Classes

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Peter Bürgisser
Keyword(s):  
Finite Fields ◽  
Structural Complexity ◽  
Polynomial Hierarchy ◽  
Complexity Classes ◽  
Np Completeness ◽  
Minimal Pairs ◽  
International Audience

International audience In Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay, Ladner, and Schöning.\par We show that if Valiant's hypothesis is true, then there is a p-definable family, which is neither p-computable nor \textitVNP-complete. More generally, we define the posets of p-degrees and c-degrees of p-definable families and prove that any countable poset can be embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for \textitVP in \textitVNP.\par Over finite fields, we give a \emphspecific example of a family of polynomials which is neither \textitVNP-complete nor p-computable, provided the polynomial hierarchy does not collapse.\par We define relativized complexity classes VP^h and VNP^h and construct complete families in these classes. Moreover, we prove that there is a p-family h satisfying VP^h = VNP^h.

scholarly journals A Parameterized Complexity View on Collapsing k-Cores

2021 ◽  
Author(s):  
Junjie Luo ◽  
Hendrik Molter ◽  
Ondřej Suchý
Keyword(s):  
Parameterized Complexity ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Structural Parameters ◽  
Vertex Cover ◽  
Polynomial Kernel ◽  
Input Graph ◽  
Induced Subgraph ◽  
Vertex Deletion ◽  
Drop Outs

AbstractWe study the -hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. (2017) and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be -hard for all r ≥ 0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r. We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k ≤ 2 and k ≥ 3. For the latter case it is known that for all x ≥ 0 Collapsed k-Core is -hard when parameterized by b. For k ≤ 2 we show that Collapsed k-Core is -hard when parameterized by b and in when parameterized by (b + x). Furthermore, we outline that Collapsed k-Core is in when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

scholarly journals On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

2021 ◽  
Vol 13 (4) ◽  
pp. 1-40
Author(s):  
Spoorthy Gunda ◽  
Pallavi Jain ◽  
Daniel Lokshtanov ◽  
Saket Saurabh ◽  
Prafullkumar Tale
Keyword(s):  
Approximation Algorithm ◽  
Arbitrary Function ◽  
Parameterized Complexity ◽  
Chordal Graphs ◽  
Polynomial Kernel ◽  
Graph Minors ◽  
Polynomial Size ◽  
Graph Operation ◽  
Approximate Polynomial ◽  
Scientific Attention

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F -Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k , F -Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and | X | ≤ k . Here, G/X is the graph obtained from G by contracting edges in X . We obtain the following results for the F - Contraction problem: • Clique Contraction is known to be FPT . However, unless NP⊆ coNP/ poly , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme ( PSAKS ). That is, it admits a (1 + ε)-approximate kernel with O ( k f(ε)) vertices for every ε > 0. • Split Contraction is known to be W[1]-Hard . We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)- FPT -approximation algorithm for Split Contraction ). Furthermore, we show that, assuming Gap-ETH , there is no (5/4-δ)- FPT -approximation algorithm for Split Contraction . Here, ε, δ > 0 are fixed constants. • Chordal Contraction is known to be W[2]-Hard . We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1] , there is no F(k) - FPT -approximation algorithm for Chordal Contraction . Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k) - FPT -approximation algorithm for the F -Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and | X | ≤ k , it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F .

scholarly journals The Parameterized Complexity of Motion Planning for Snake-Like Robots

2020 ◽  
Vol 69 ◽  
pp. 191-229
Author(s):  
Siddharth Gupta ◽  
Guy Sa'ar ◽  
Meirav Zehavi
Keyword(s):  
Motion Planning ◽  
Video Game ◽  
Parameterized Complexity ◽  
Initial Position ◽  
Polynomial Kernel ◽  
Final Position ◽  
Grid Graphs ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
General Graphs

We study the parameterized complexity of a variant of the classic video game Snake that models real-world problems of motion planning. Given a snake-like robot with an initial position and a final position in an environment (modeled by a graph), our objective is to determine whether the robot can reach the final position from the initial position without intersecting itself. Naturally, this problem models a wide-variety of scenarios, ranging from the transportation of linked wagons towed by a locomotor at an airport or a supermarket to the movement of a group of agents that travel in an “ant-like” fashion and the construction of trains in amusement parks. Unfortunately, already on grid graphs, this problem is PSPACE-complete. Nevertheless, we prove that even on general graphs, the problem is solvable in FPT time with respect to the size of the snake. In particular, this shows that the problem is fixed-parameter tractable (FPT). Towards this, we show how to employ color-coding to sparsify the configuration graph of the problem to reduce its size significantly. We believe that our approach will find other applications in motion planning. Additionally, we show that the problem is unlikely to admit a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion planning problems (where the intermediate configurations of the motion are of importance) has so far been largely overlooked. Thus, our work is pioneering in this regard.

Method application of optimal multi-parameter analysis to assess the distribution of water masses as an example of CTD data of the Barents Sea in summer 2017

2019 ◽  
pp. 38-51
Author(s):  
Valeriy G. Yakubenko ◽  
Anna L. Chultsova
Keyword(s):  
Barents Sea ◽  
Field Measurements ◽  
Structural Similarity ◽  
Water Masses ◽  
Parameter Analysis ◽  
Water Dynamics ◽  
Phosphorus Concentration ◽  
Nitrogen And Phosphorus ◽  
The Barents Sea ◽  
Expert Assessments

Identification of water masses in areas with complex water dynamics is a complex task, which is usually solved by the method of expert assessments. In this paper, it is proposed to use a formal procedure based on the application of the method of optimal multiparametric analysis (OMP analysis). The data of field measurements obtained in the 68th cruise of the R/V “Academician Mstislav Keldysh” in the summer of 2017 in the Barents Sea on the distribution of temperature, salinity, oxygen, silicates, nitrogen, and phosphorus concentration are used as a data for research. A comparison of the results with data on the distribution of water masses in literature based on expert assessments (Oziel et al., 2017), allows us to conclude about their close structural similarity. Some differences are related to spatial and temporal shifts of measurements. This indicates the feasibility of using the OMP analysis technique in oceanological studies to obtain quantitative data on the spatial distribution of different water masses.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

Conclusion

2018 ◽  
Author(s):  
Alistair Fox
Keyword(s):  
National Identity ◽  
New Zealand ◽  
Identity Formation ◽  
Essential Role ◽  
Coming Of Age ◽  
Financial Returns ◽  
Self Recognition ◽  
International Audience ◽  
National Identity Formation

The conclusion reaffirms the essential role played by cinema generally, and the coming-of-age genre in particular, in the process of national identity formation, because of its effectiveness in facilitating self-recognition and self-experience through a process of triangulation made possible, for the most part, by a dialogue with some of the nation’s most iconic works of literature. This section concludes by point out the danger posed, however, by an observable trend toward generic standardization in New Zealand films motivated by a desire to appeal to an international audience out of consideration for the financial returns expected by funding bodies under current regimes.

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