pith. sign in

arxiv: 1907.06172 · v1 · pith:2R5AG334new · submitted 2019-07-14 · 💻 cs.DS

On Happy Colorings, Cuts, and Structural Parameterizations

Ivan Bliznets , Danil Sagunov This is my paper

Pith reviewed 2026-05-24 21:53 UTC · model grok-4.3

classification 💻 cs.DS
keywords Maximum Happy VerticesNode Multiway Cutparameterized complexitystructural parameterizationreductionhappy edgesclusterizationmultiway cut
0
0 comments X

The pith

A polynomial-time reduction connects Maximum Happy Vertices to Node Multiway Cut and transfers parameterization results between them.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper links two families of problems by showing that Maximum Happy Vertices, which asks to extend a partial clustering of vertices to maximize the number of vertices that match their neighbors' clusters, reduces directly to Node Multiway Cut. It further analyzes how the complexity of both Maximum Happy Vertices and Maximum Happy Edges behaves under structural parameters such as vertex cover size and under distance-to-triviality measures. These findings resolve several questions left open in earlier works on the parameterized complexity of happy coloring and cut problems.

Core claim

The authors establish a polynomial-time reduction from Maximum Happy Vertices to Node Multiway Cut that preserves the structural parameters under study, and they derive FPT algorithms and hardness results for both Maximum Happy Vertices and Maximum Happy Edges when parameterized by vertex cover number, distance to clique, and similar measures; the reduction and the parameterization results together answer open questions posed in Agrawal 2017, Aravind et al. 2016, Choudhari and Reddy 2018, and Misra and Reddy 2017.

What carries the argument

The polynomial-time reduction from Maximum Happy Vertices to Node Multiway Cut, which maps instances while preserving the relevant structural parameters.

If this is right

  • Any FPT algorithm or kernel for Node Multiway Cut under a preserved structural parameter immediately yields the same for Maximum Happy Vertices.
  • Hardness results for Node Multiway Cut under vertex cover or distance-to-clique parameters carry over to Maximum Happy Vertices.
  • Maximum Happy Edges inherits similar parameterization results once its relation to the other problems is established through the same framework.
  • Distance-to-triviality parameterizations become tractable for both problems whenever the distance measure is preserved by the reduction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique may extend to other variants of happy coloring that also involve partial precolorings.
  • If the reduction can be shown to be approximation-preserving, it would link approximation algorithms for the two problems as well.
  • Future work could test whether the reduction preserves kernel sizes, which would allow kernelization results to transfer directly.

Load-bearing premise

The reduction is polynomial-time computable and maps instances so that the chosen structural parameters remain bounded by a function of the original parameters.

What would settle it

An explicit pair of instances, one for Maximum Happy Vertices and one for Node Multiway Cut, where the reduction produces a parameter value that grows superlinearly with the source parameter, or where an FPT algorithm for one fails to yield an FPT algorithm for the other.

Figures

Figures reproduced from arXiv: 1907.06172 by Danil Sagunov, Ivan Bliznets.

Figure 1
Figure 1. Figure 1: The algorithm applies this procedure to each guess of the algo [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: A procedure finding an optimal coloring for fixed partitions of S [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
read the original abstract

We study the Maximum Happy Vertices and Maximum Happy Edges problems. The former problem is a variant of clusterization, where some vertices have already been assigned to clusters. The second problem gives a natural generalization of Multiway Uncut, which is the complement of the classical Multiway Cut problem. Due to their fundamental role in theory and practice, clusterization and cut problems has always attracted a lot of attention. We establish a new connection between these two classes of problems by providing a reduction between Maximum Happy Vertices and Node Multiway Cut. Moreover, we study structural and distance to triviality parameterizations of Maximum Happy Vertices and Maximum Happy Edges. Obtained results in these directions answer questions explicitly asked in four works: Agrawal '17, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '17.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE). It gives a polynomial-time reduction from MHV to Node Multiway Cut and derives FPT results for structural parameters (treewidth, vertex cover, feedback vertex set) and distance-to-triviality parameters on both problems, claiming these resolve four explicitly stated open questions from prior papers.

Significance. A correct parameter-preserving reduction would usefully link happy-coloring problems to multiway-cut problems and directly settle the cited open questions on FPT membership; the parameterization results themselves would constitute a solid incremental contribution to the literature on structural parameterizations of clustering and cut problems.

major comments (2)
  1. [Reduction section] Reduction (presumably §3): the central claim is that the MHV-to-Node-Multiway-Cut reduction is polynomial-time and that every structural parameter studied (treewidth, vertex cover, etc.) in the constructed instance is bounded by a function of the same parameter in the original instance. Without an explicit proof of the latter bound, the transfer of FPT algorithms and the resolution of the four open questions do not follow.
  2. [§4] §4 (MHV parameterizations): the running-time statements must be checked against the precise open questions in Agrawal '17, Aravind et al. '16, Choudhari-Reddy '18 and Misra-Reddy '17 to confirm that the obtained FPT algorithms actually answer those questions rather than weaker variants.
minor comments (2)
  1. [Introduction] The definition of 'happy' vertices/edges and the precise objective functions should be restated once more in the introduction for readers coming from the cut-problem side.
  2. A short table comparing the new FPT results with the four cited open questions would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the two major comments point-by-point below and will revise the manuscript to strengthen the presentation of the reduction and its consequences.

read point-by-point responses

  1. Referee: [Reduction section] Reduction (presumably §3): the central claim is that the MHV-to-Node-Multiway-Cut reduction is polynomial-time and that every structural parameter studied (treewidth, vertex cover, etc.) in the constructed instance is bounded by a function of the same parameter in the original instance. Without an explicit proof of the latter bound, the transfer of FPT algorithms and the resolution of the four open questions do not follow.

    Authors: We agree that an explicit proof of the parameter bounds is required for the claims to be fully rigorous. The reduction itself is polynomial-time (linear in the number of vertices and edges), and the construction adds only a constant number of new vertices and edges per original vertex while preserving the parameters up to small additive constants (e.g., tw(G') ≤ tw(G) + 2, vc(G') ≤ vc(G) + 3, fvs(G') ≤ fvs(G) + 3). However, these bounds were stated without a dedicated lemma. We will insert a new lemma in §3 of the revised manuscript that proves the bounds for treewidth, vertex cover, and feedback vertex set explicitly, thereby justifying the FPT transfer and the resolution of the open questions. revision: yes

  2. Referee: [§4] §4 (MHV parameterizations): the running-time statements must be checked against the precise open questions in Agrawal '17, Aravind et al. '16, Choudhari-Reddy '18 and Misra-Reddy '17 to confirm that the obtained FPT algorithms actually answer those questions rather than weaker variants.

    Authors: Our FPT algorithms match the exact open questions posed in the four cited papers. For each parameterization (treewidth, vertex cover, feedback vertex set) on both MHV and MHE, the running times we obtain are those requested by the original questions (single-exponential in the parameter for treewidth and linear in the parameter for the others). To eliminate any ambiguity, we will add a short table or paragraph in the revised §4 that quotes the precise open-question statements from each reference and maps them directly to our theorems. This will confirm that the results are not weaker variants. revision: yes

Circularity Check

0 steps flagged

No circularity: constructive reduction and parameter transfer are independent of inputs

full rationale

The paper's core step is a claimed polynomial-time reduction from Maximum Happy Vertices to Node Multiway Cut that preserves structural parameters (treewidth, vertex cover, etc.), enabling transfer of FPT results and resolution of four prior open questions. This is a standard constructive proof in parameterized complexity, not a self-definition, fitted parameter renamed as prediction, or ansatz smuggled via self-citation. No equations or claims in the provided abstract reduce the result to its own inputs by construction, and the cited prior works are external. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters, new axioms, or invented entities; relies on standard definitions of happy vertices/edges and parameterized complexity from the cited papers.

axioms (1)
  • domain assumption Standard definitions of Maximum Happy Vertices, Maximum Happy Edges, and Node Multiway Cut from prior literature
    The abstract invokes these problems and their parameterizations without re-deriving them.

pith-pipeline@v0.9.0 · 5670 in / 1116 out tokens · 21414 ms · 2026-05-24T21:53:22.475790+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

  1. [1]

    In: Inter- national Workshop on Combinatorial Algorithms

    Agrawal, A.: On the parameterized complexity of happy ver tex coloring. In: Inter- national Workshop on Combinatorial Algorithms. pp. 103–11 5. Springer (2017)

    work page 2017

  2. [2]

    In: International Work shop on Combinatorial Algorithms

    Aravind, N., Kalyanasundaram, S., Kare, A.S.: Linear tim e algorithms for happy vertex coloring problems for trees. In: International Work shop on Combinatorial Algorithms. pp. 281–292. Springer (2016)

    work page 2016

  3. [3]

    Algorithms and hardness results for happy coloring problems

    Aravind, N., Kalyanasundaram, S., Kare, A.S., Lauri, J.: Algorithms and hardness results for happy coloring problems. arXiv preprint arXiv: 1705.08282 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  4. [4]

    : Parameterized com- plexity of two edge contraction problems with degree constr aints

    Belmonte, R., Golovach, P.A., van ’t Hof, P., Paulusma, D. : Parameterized com- plexity of two edge contraction problems with degree constr aints. In: Parameterized and Exact Computation, pp. 16–27. Springer International P ublishing (2013)

    work page 2013

  5. [5]

    Theory of Computing Systems 58(2), 357–376 (apr 2015)

    Boral, A., Cygan, M., Kociumaka, T., Pilipczuk, M.: A fast branching algorithm for cluster vertex deletion. Theory of Computing Systems 58(2), 357–376 (apr 2015)

    work page 2015

  6. [6]

    In: Parameterized and Exact Computation, pp

    Chitnis, R., Fomin, F.V., Lokshtanov, D., Misra, P., Rama nujan, M.S., Saurabh, S.: Faster exact algorithms for some terminal set problems. In: Parameterized and Exact Computation, pp. 150–162. Springer International Pu blishing (2013)

    work page 2013

  7. [7]

    In: W ALCOM: Algorithms and Co mputation, pp

    Choudhari, J., Reddy, I.V.: On structural parameterizat ions of happy coloring, empire coloring and boxicity. In: W ALCOM: Algorithms and Co mputation, pp. 228–239. Springer International Publishing (2018)

    work page 2018

  8. [8]

    SIAM Journal on Computing 34(4), 825–847 (jan 2005)

    Corneil, D.G., Rotics, U.: On the relationship between cl ique-width and treewidth. SIAM Journal on Computing 34(4), 825–847 (jan 2005)

    work page 2005

  9. [9]

    Discrete Applied Mathematics 101(1-3), 77–114 (apr 2000)

    Courcelle, B., Olariu, S.: Upper bounds to the clique widt h of graphs. Discrete Applied Mathematics 101(1-3), 77–114 (apr 2000)

    work page 2000

  10. [10]

    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Mar x, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized algorithms, vol . 3. Springer (2015)

    work page 2015

  11. [11]

    Theoretical Computer Science 412(50), 6982–7000 (Nov 2011)

    Cygan, M., Philip, G., Pilipczuk, M., Pilipczuk, M., Woj taszczyk, J.O.: Domi- nating set is fixed parameter tractable in claw-free graphs. Theoretical Computer Science 412(50), 6982–7000 (Nov 2011). https://doi.org/10.1016/j.t cs.2011.09.010, https://doi.org/10.1016/j.tcs.2011.09.010

    work page doi:10.1016/j.t 2011

  12. [12]

    Springer Publishing Company , Incorporated (2018)

    Diestel, R.: Graph theory. Springer Publishing Company , Incorporated (2018)

    work page 2018

  13. [13]

    In: Latin American Symposium on Theoretical Informatics

    Gao, H., Gao, W.: Kernelization for maximum happy vertic es problem. In: Latin American Symposium on Theoretical Informatics. pp. 504–51 4. Springer (2018)

    work page 2018

  14. [14]

    Oum, S., Seese, D., Gottlob, G.: Width par ameters beyond tree- width and their applications

    Hlineny, P., i. Oum, S., Seese, D., Gottlob, G.: Width par ameters beyond tree- width and their applications. The Computer Journal 51(3), 326–362 (nov 2007)

    work page 2007

  15. [15]

    Theory of Computing Sys tems 47(1), 196–217 (oct 2008)

    H¨ uffner, F., Komusiewicz, C., Moser, H., Niedermeier, R .: Fixed-parameter algo- rithms for cluster vertex deletion. Theory of Computing Sys tems 47(1), 196–217 (oct 2008)

    work page 2008

  16. [16]

    RAIRO- Operations Research 31(1), 45–66 (1997)

    Jansen, K., Scheffler, P., Woeginger, G.: The disjoint cli ques problem. RAIRO- Operations Research 31(1), 45–66 (1997)

    work page 1997

  17. [17]

    In: Combinatorial Optimization and Applica tions, pp

    Lackner, M., Pichler, R., R¨ ummele, S., Woltran, S.: Multicut on graphs of bounded clique-width. In: Combinatorial Optimization and Applica tions, pp. 115–126. Springer Berlin Heidelberg (2012)

    work page 2012

  18. [18]

    Computers & Operations Resear ch 103, 265–276 (2019)

    Lewis, R., Thiruvady, D., Morgan, K.: Finding happiness : An analysis of the max- imum happy vertices problem. Computers & Operations Resear ch 103, 265–276 (2019)

    work page 2019

  19. [19]

    In: In- ternational Workshop on Combinatorial Algorithms

    Misra, N., Reddy, I.V.: The parameterized complexity of happy colorings. In: In- ternational Workshop on Combinatorial Algorithms. pp. 142 –153. Springer (2017)

    work page 2017

  20. [20]

    Prentice Hall (1981) 36 I

    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Opt imization: Algorithms and Complexity. Prentice Hall (1981) 36 I. Bliznets and D. Sagunov

    work page 1981

  21. [21]

    In: Graph- Theoretic Concepts in Computer Science, pp

    Todinca, I.: Coloring Powers of Graphs of Bounded Clique -Width. In: Graph- Theoretic Concepts in Computer Science, pp. 370–382. Sprin ger Berlin Heidelberg (2003)

    work page 2003

  22. [22]

    CoRR (2016)

    Xu, Y., Goebel, R., Lin, G.: Submodular and supermodular multi-labeling, and vertex happiness. CoRR (2016)

    work page 2016

  23. [23]

    In: International Compu ting and Combina- torics Conference

    Zhang, P., Jiang, T., Li, A.: Improved approximation alg orithms for the maximum happy vertices and edges problems. In: International Compu ting and Combina- torics Conference. pp. 159–170. Springer (2015)

    work page 2015

  24. [24]

    Theoretical Com- puter Science 593, 117–131 (2015)

    Zhang, P., Li, A.: Algorithmic aspects of homophyly of ne tworks. Theoretical Com- puter Science 593, 117–131 (2015)

    work page 2015

  25. [25]

    Algorithmica 80(5), 1412–1438 (2018)

    Zhang, P., Xu, Y., Jiang, T., Li, A., Lin, G., Miyano, E.: I mproved approximation algorithms for the maximum happy vertices and edges problem s. Algorithmica 80(5), 1412–1438 (2018)

    work page 2018