TY - JOUR A1 - Casel, Katrin A1 - Fischbeck, Philipp A1 - Friedrich, Tobias A1 - Göbel, Andreas A1 - Lagodzinski, J. A. Gregor T1 - Zeros and approximations of Holant polynomials on the complex plane JF - Computational complexity : CC N2 - We present fully polynomial time approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures known as polymers in statistical physics. Our method involves establishing zero-free regions for the partition functions of polymer models and using the most significant terms of the cluster expansion to approximate them. Results of our technique include new approximation and sampling algorithms for a diverse class of Holant polynomials in the low-temperature regime (i.e. small external field) and approximation algorithms for general Holant problems with small signature weights. Additionally, we give randomised approximation and sampling algorithms with faster running times for more restrictive classes. Finally, we improve the known zero-free regions for a perfect matching polynomial. KW - Holant problems KW - approximate counting KW - partition functions KW - graph KW - polynomials Y1 - 2022 U6 - https://doi.org/10.1007/s00037-022-00226-5 SN - 1016-3328 SN - 1420-8954 VL - 31 IS - 2 PB - Springer CY - Basel ER - TY - JOUR A1 - Casel, Katrin A1 - Fernau, Henning A1 - Gaspers, Serge A1 - Gras, Benjamin A1 - Schmid, Markus L. T1 - On the complexity of the smallest grammar problem over fixed alphabets JF - Theory of computing systems N2 - In the smallest grammar problem, we are given a word w and we want to compute a preferably small context-free grammar G for the singleton language {w} (where the size of a grammar is the sum of the sizes of its rules, and the size of a rule is measured by the length of its right side). It is known that, for unbounded alphabets, the decision variant of this problem is NP-hard and the optimisation variant does not allow a polynomial-time approximation scheme, unless P = NP. We settle the long-standing open problem whether these hardness results also hold for the more realistic case of a constant-size alphabet. More precisely, it is shown that the smallest grammar problem remains NP-complete (and its optimisation version is APX-hard), even if the alphabet is fixed and has size of at least 17. The corresponding reduction is robust in the sense that it also works for an alternative size-measure of grammars that is commonly used in the literature (i. e., a size measure also taking the number of rules into account), and it also allows to conclude that even computing the number of rules required by a smallest grammar is a hard problem. On the other hand, if the number of nonterminals (or, equivalently, the number of rules) is bounded by a constant, then the smallest grammar problem can be solved in polynomial time, which is shown by encoding it as a problem on graphs with interval structure. However, treating the number of rules as a parameter (in terms of parameterised complexity) yields W[1]-hardness. Furthermore, we present an O(3(vertical bar w vertical bar)) exact exponential-time algorithm, based on dynamic programming. These three main questions are also investigated for 1-level grammars, i. e., grammars for which only the start rule contains nonterminals on the right side; thus, investigating the impact of the "hierarchical depth" of grammars on the complexity of the smallest grammar problem. In this regard, we obtain for 1-level grammars similar, but slightly stronger results. KW - grammar-based compression KW - smallest grammar problem KW - straight-line KW - programs KW - NP-completeness KW - exact exponential-time algorithms Y1 - 2020 U6 - https://doi.org/10.1007/s00224-020-10013-w SN - 1432-4350 SN - 1433-0490 VL - 65 IS - 2 SP - 344 EP - 409 PB - Springer CY - New York ER - TY - JOUR A1 - Casel, Katrin A1 - Dreier, Jan A1 - Fernau, Henning A1 - Gobbert, Moritz A1 - Kuinke, Philipp A1 - Villaamil, Fernando Sánchez A1 - Schmid, Markus L. A1 - van Leeuwen, Erik Jan T1 - Complexity of independency and cliquy trees JF - Discrete applied mathematics N2 - An independency (cliquy) tree of an n-vertex graph G is a spanning tree of G in which the set of leaves induces an independent set (clique). We study the problems of minimizing or maximizing the number of leaves of such trees, and fully characterize their parameterized complexity. We show that all four variants of deciding if an independency/cliquy tree with at least/most l leaves exists parameterized by l are either Para-NP- or W[1]-hard. We prove that minimizing the number of leaves of a cliquy tree parameterized by the number of internal vertices is Para-NP-hard too. However, we show that minimizing the number of leaves of an independency tree parameterized by the number k of internal vertices has an O*(4(k))-time algorithm and a 2k vertex kernel. Moreover, we prove that maximizing the number of leaves of an independency/cliquy tree parameterized by the number k of internal vertices both have an O*(18(k))-time algorithm and an O(k 2(k)) vertex kernel, but no polynomial kernel unless the polynomial hierarchy collapses to the third level. Finally, we present an O(3(n) . f(n))-time algorithm to find a spanning tree where the leaf set has a property that can be decided in f (n) time and has minimum or maximum size. KW - independency tree KW - cliquy tree KW - parameterized complexity KW - Kernelization KW - algorithms KW - exact algorithms Y1 - 2018 U6 - https://doi.org/10.1016/j.dam.2018.08.011 SN - 0166-218X SN - 1872-6771 VL - 272 SP - 2 EP - 15 PB - Elsevier CY - Amsterdam [u.a.] ER -