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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

For graphs G and H, an H-coloring of G is an edge-preserving mapping from V(G) to V(H). In the H-Coloring problem the graph H is fixed and we ask whether an instance graph G admits an H-coloring. A generalization of this problem is H-ColoringExt, where some vertices of G are already mapped to vertices of H and we ask if this partial mapping can be extended to an H-coloring.
We study the complexity of variants of H-Coloring in F-free graphs, i.e., graphs excluding a fixed graph F as an induced subgraph. For integers a,b,c ⩾ 1, by S_{a,b,c} we denote the graph obtained by identifying one endvertex of three paths on a+1, b+1, and c+1 vertices, respectively. For odd k ⩾ 5, by W_k we denote the graph obtained from the k-cycle by adding a universal vertex.
As our main algorithmic result we show that W_5-ColoringExt is polynomial-time solvable in S_{2,1,1}-free graphs. This result exhibits an interesting non-monotonicity of H-ColoringExt with respect to taking induced subgraphs of H. Indeed, W_5 contains a triangle, and K_3-Coloring, i.e., classical 3-coloring, is NP-hard already in claw-free (i.e., S_{1,1,1}-free) graphs. Our algorithm is based on two main observations:
1) W_5-ColoringExt in S_{2,1,1}-free graphs can be in polynomial time reduced to a variant of the problem of finding an independent set intersecting all triangles, and
2) the latter problem can be solved in polynomial time in S_{2,1,1}-free graphs.
We complement this algorithmic result with several negative ones. In particular, we show that W_5-Coloring is NP-hard in P_t-free graphs for some constant t and W_5-ColoringExt is NP-hard in S_{3,3,3}-free graphs of bounded degree. This is again uncommon, as usually problems that are NP-hard in S_{a,b,c}-free graphs for some constant a,b,c are already hard in claw-free graphs

Michał Dębski, Zbigniew Lonc, Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Computing Homomorphisms in Hereditary Graph Classes: The Peculiar Case of the 5-Wheel and Graphs with No Long Claws. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{debski_et_al:LIPIcs.ISAAC.2022.14, author = {D\k{e}bski, Micha{\l} and Lonc, Zbigniew and Okrasa, Karolina and Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Computing Homomorphisms in Hereditary Graph Classes: The Peculiar Case of the 5-Wheel and Graphs with No Long Claws}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {14:1--14:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.14}, URN = {urn:nbn:de:0030-drops-172996}, doi = {10.4230/LIPIcs.ISAAC.2022.14}, annote = {Keywords: graph homomorphism, forbidden induced subgraphs, precoloring extension} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We study the 3-Coloring problem in graphs with small diameter. In 2013, Mertzios and Spirakis showed that for n-vertex diameter-2 graphs this problem can be solved in subexponential time 2^{𝒪(√{n log n})}. Whether the problem can be solved in polynomial time remains a well-known open question in the area of algorithmic graphs theory.
In this paper we present an algorithm that solves 3-Coloring in n-vertex diameter-2 graphs in time 2^{𝒪(n^{1/3} log² n)}. This is the first improvement upon the algorithm of Mertzios and Spirakis in the general case, i.e., without putting any further restrictions on the instance graph.
In addition to standard branchings and reducing the problem to an instance of 2-Sat, the crucial building block of our algorithm is a combinatorial observation about 3-colorable diameter-2 graphs, which is proven using a probabilistic argument.
As a side result, we show that 3-Coloring can be solved in time 2^{𝒪((n log n)^{2/3})} in n-vertex diameter-3 graphs. We also generalize our algorithms to the problem of finding a list homomorphism from a small-diameter graph to a cycle.

Michał Dębski, Marta Piecyk, and Paweł Rzążewski. Faster 3-Coloring of Small-Diameter Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{debski_et_al:LIPIcs.ESA.2021.37, author = {D\k{e}bski, Micha{\l} and Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Faster 3-Coloring of Small-Diameter Graphs}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {37:1--37:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.37}, URN = {urn:nbn:de:0030-drops-146181}, doi = {10.4230/LIPIcs.ESA.2021.37}, annote = {Keywords: 3-coloring, fine-grained complexity, subexponential-time algorithm, diameter} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

For graphs G,H, a homomorphism from G to H is an edge-preserving mapping from V(G) to V(H). In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is equipped with a list L(v) ⊆ V(H), and we need to determine whether there exists a homomorphism from G to H which additionally respects the lists L. List homomorphisms are a natural generalization of (list) colorings.
Very recently Okrasa, Piecyk, and Rzążewski [ESA 2020] studied the fine-grained complexity of the problem, parameterized by the treewidth of the instance graph G. They defined a new invariant i^*(H), and proved that for every relevant graph H, i.e., such that LHom(H) is NP-hard, this invariant is the correct base of the exponent in the running time of any algorithm solving the LHom(H) problem.
In this paper we continue this direction and study the complexity of the problem under different parameterizations. As the first result, we show that i^*(H) is also the right complexity base if the parameter is the size of a minimum feedback vertex set of G, denoted by fvs(G). In particular, for every relevant graph H, the LHom(H) problem
- can be solved in time i^*(H)^fvs(G) ⋅ |V(G)|^𝒪(1), if a minimum feedback vertex set of G is given,
- cannot be solved in time (i^*(H) - ε)^fvs(G) ⋅ |V(G)|^𝒪(1), for any ε > 0, unless the SETH fails. Then we turn our attention to a parameterization by the cutwidth ctw(G) of G. Jansen and Nederlof [TCS 2019] showed that List k-Coloring (i.e., LHom(K_k)) can be solved in time c^ctw(G) ⋅ |V(G)|^𝒪(1) for an absolute constant c, i.e., the base of the exponential function does not depend on the number of colors. Jansen asked whether this behavior extends to graph homomorphisms. As the main result of the paper, we answer the question in the negative. We define a new graph invariant mim^*(H), closely related to the size of a maximum induced matching in H, and prove that for all relevant graphs H, the LHom(H) problem cannot be solved in time (mim^*(H)-ε)^{ctw(G)}⋅ |V(G)|^𝒪(1) for any ε > 0, unless the SETH fails. In particular, this implies that, assuming the SETH, there is no constant c, such that for every odd cycle the non-list version of the problem can be solved in time c^ctw(G) ⋅ |V(G)|^𝒪(1).

Marta Piecyk and Paweł Rzążewski. Fine-Grained Complexity of the List Homomorphism Problem: Feedback Vertex Set and Cutwidth. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 56:1-56:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{piecyk_et_al:LIPIcs.STACS.2021.56, author = {Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Fine-Grained Complexity of the List Homomorphism Problem: Feedback Vertex Set and Cutwidth}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {56:1--56:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.56}, URN = {urn:nbn:de:0030-drops-137012}, doi = {10.4230/LIPIcs.STACS.2021.56}, annote = {Keywords: list homomorphisms, fine-grained complexity, SETH, feedback vertex set, cutwidth} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is assigned with a list L(v) of vertices of H. We ask whether there exists a homomorphism h from G to H, which respects lists L, i.e., for every v ∈ V(G) it holds that h(v) ∈ L(v).
The complexity dichotomy for LHom(H) was proven by Feder, Hell, and Huang [JGT 2003]. The authors showed that the problem is polynomial-time solvable if H belongs to the class called bi-arc graphs, and for all other graphs H it is NP-complete.
We are interested in the complexity of the LHom(H) problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rzążewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H, i.e., if every vertex has a loop.
In this paper we extend and generalize their results for all relevant graphs H, i.e., those, for which the LHom(H) problem is NP-hard. For every such H we find a constant k = k(H), such that the LHom(H) problem on instances G with n vertices and treewidth t
- can be solved in time k^t ⋅ n^𝒪(1), provided that G is given along with a tree decomposition of width t,
- cannot be solved in time (k-ε)^t ⋅ n^𝒪(1), for any ε > 0, unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)|.
Obtaining matching upper and lower bounds shows that the set of algorithmic tools that we have discovered cannot be extended in order to obtain faster algorithms for LHom(H) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of the LHom(H) problem, e.g. with different parameterizations.

Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 74:1-74:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{okrasa_et_al:LIPIcs.ESA.2020.74, author = {Okrasa, Karolina and Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {74:1--74:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.74}, URN = {urn:nbn:de:0030-drops-129402}, doi = {10.4230/LIPIcs.ESA.2020.74}, annote = {Keywords: list homomorphisms, fine-grained complexity, SETH, treewidth} }

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