10 research outputs found
Approximating Highly Inapproximable Problems on Graphs of Bounded Twin-Width
For any , we give a polynomial-time
-approximation algorithm for Max Independent Set in graphs of
bounded twin-width given with an -sequence. This result is derived from
the following time-approximation trade-off: We establish an
-approximation algorithm running in time ,
for every integer . Guided by the same framework, we obtain
similar approximation algorithms for Min Coloring and Max Induced Matching. In
general graphs, all these problems are known to be highly inapproximable: for
any , a polynomial-time -approximation for
any of them would imply that PNP [Hastad, FOCS '96; Zuckerman, ToC '07;
Chalermsook et al., SODA '13]. We generalize the algorithms for Max Independent
Set and Max Induced Matching to the independent (induced) packing of any fixed
connected graph . In contrast, we show that such approximation guarantees on
graphs of bounded twin-width given with an -sequence are very unlikely
for Min Independent Dominating Set, and somewhat unlikely for Longest Path and
Longest Induced Path. Regarding the existence of better approximation
algorithms, there is a (very) light evidence that the obtained approximation
factor of for Max Independent Set may be best possible. This is
the first in-depth study of the approximability of problems in graphs of
bounded twin-width. Prior to this paper, essentially the only such result was
a~polynomial-time -approximation algorithm for Min Dominating Set [Bonnet
et al., ICALP '21].Comment: 30 pages, 3 figures, 1 tabl
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
A graph is -free if it does not contain pairwise vertex-disjoint and
non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in
-free graphs can be solved in quasi-polynomial time. As a main technical
result, we establish that "sparse" (here, not containing large complete
bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback
vertex set number) at most logarithmic in the number of vertices. This is
proven sharp as there is an infinite family of -free graphs without
-subgraph and whose treewidth is (at least) logarithmic.
Other consequences include that most of the central NP-complete problems
(such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set,
Minimum Coloring) can be solved in polynomial time in sparse -free graphs,
and that deciding the -freeness of sparse graphs is polynomial time
solvable.Comment: 28 pages, 6 figures. v3: improved complexity result
Suite de permutations lors d'une course de n coureurs de vitesses constantes
On considĂšre n coureurs avec des vitesses constantes et diĂ©rentes sur une piste circulaire de longueur unitaire. On les numĂ©rote de 1 Ă n ; leur ordre sur la piste forme une permutation.On Ă©tudie la suite de ces permutations au cours du temps et plus particuliĂšrement, combien de permutations apparraissent. On utilise le thĂ©orĂšme de Kronecker pour montrer qu'avec des vitesses Q linĂ©airement indĂ©pendantes, toutes les permutations apparraissent. On montre Ă©galement que, dans ce cas, les frĂ©quences d'apparition sont les mĂȘmes. Plus gĂ©nĂ©ralement on donne une mĂ©thode gĂ©omĂ©trique pour calculer la frĂ©quence dans un cas quelconque
Deciding twin-width at most 4 is NP-complete
International audienceWe show that determining if an n-vertex graph has twin-width at most 4 is NP-complete, and requires time 2 âŠ(n/ log n) unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that n-vertex graphs subdivided at least 2 log n times have twin-width at most 4. We also show how to encode trigraphs H (2-edge colored graphs involved in the definition of twin-width) into graphs G, in the sense that every d-sequence (sequence of vertex contractions witnessing that the twin-width is at most d) of G inevitably creates H as an induced subtrigraph, whereas there exists a partial d-sequence that actually goes from G to H. We believe that these facts and their proofs can be of independent interest
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
27 pages, 6 figuresA graph is -free if it does not contain pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices, which is sharp already for . As a consequence, most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in these graphs, and in particular deciding the -freeness of sparse graphs is polytime
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
27 pages, 6 figuresA graph is -free if it does not contain pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices, which is sharp already for . As a consequence, most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in these graphs, and in particular deciding the -freeness of sparse graphs is polytime
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
27 pages, 6 figuresA graph is -free if it does not contain pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices, which is sharp already for . As a consequence, most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in these graphs, and in particular deciding the -freeness of sparse graphs is polytime