Complexity Dichotomy Research Articles

Dichotomies for tree minor containment with structural parameters

The problem of determining whether a graph G contains another graph H as a minor, referred to as the minor containment problem, is a fundamental problem in the field of graph algorithms. While the problem is NP-complete in general, it can be tractable on some restricted graph classes. This study focuses on the case where both G and H are trees, known as the tree minor containment problem. Even in this case, the problem is known to be NP-complete. In contrast, polynomial-time algorithms are known for the case when both trees are caterpillars or when the maximum degree of H is a constant. Our research aims to clarify the boundary of tractability and intractability for the tree minor containment problem. Specifically, we provide complexity dichotomies for the problem based on three structural parameters: diameter, pathwidth, and path eccentricity.

Unit-length Rectangular Drawings of Graphs

A rectangular drawing of a planar graph $G$ is a planar drawing of $G$ in which vertices are mapped to grid points, edges are mapped to horizontal and vertical straight-line segments, and faces are drawn as rectangles. Sometimes this latter constraint is relaxed for the outer face. In this paper, we study rectangular drawings in which the edges have unit length. We show a complexity dichotomy for the problem of deciding the existence of a unit-length rectangular drawing, depending on whether the outer face must also be drawn as a rectangle or not. Specifically, we prove that the problem is NP-complete for biconnected graphs when the drawing of the outer face is not required to be a rectangle, even if the sought drawing must respect a given combinatorial embedding, whereas it is polynomial-time solvable, both in the fixed and the variable embedding settings, if the outer face is required to be drawn as a rectangle. Furthermore, we provide a linear-time algorithm for deciding whether a plane graph admits an embedding-preserving unit-length rectangular drawing if the drawing of the outer face is prescribed. As a by-product of our research, we provide the first polynomial-time algorithm to test whether a planar graph $G$ admits a rectangular drawing, for general instances of maximum degree $4$.

Restricted holant dichotomy on domain sizes 3 and 4

Holant⁎(f) denotes a class of counting problems specified by a constraint function f. We prove complexity dichotomy theorems for Holant⁎(f) in two settings: (1) f is any symmetric arity-3 real-valued function on input of domain size 3. (2) f is any symmetric arity-3 {0,1}-valued function on input of domain size 4.

On the complexity of distance-d independent set reconfiguration

For a fixed positive integer d≥2, a distance-d independent set (DdIS) of a graph is a vertex-subset whose distance between any two members is at least d. Imagine that there is a token placed on each member of a DdIS. Two DdISs are adjacent under Token Sliding (TS) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping (TJ), the target vertex needs not to be adjacent to the original one. The Distance-dIndependent Set Reconfiguration (DdISR) problem under TS/TJ asks if there is a corresponding sequence of adjacent DdISs that transforms one given DdIS into another. The problem for d=2, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of DdISR on different graphs under TS and TJ for any fixed d≥3. On chordal graphs, we show that DdISR under TJ is in P when d is even and PSPACE-complete when d is odd. On split graphs, there is an interesting complexity dichotomy: DdISR is PSPACE-complete for d=2 but in P for d=3 under TS, while under TJ it is in P for d=2 but PSPACE-complete for d=3. Additionally, certain well-known hardness results for d=2 on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for d≥3.

A Complexity Dichotomy in Spatial Reasoning via Ramsey Theory

Constraint satisfaction problems (CSPs) for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to obtain polynomial-time tractability results for such CSPs is a certain reduction to polynomial-time tractable finite-domain CSPs defined over k -types, for a sufficiently large k . We give sufficient conditions when this method can be applied and apply these conditions to obtain a new complexity dichotomy for CSPs of first-order expansions of the basic relations of the well-studied spatial reasoning formalism RCC5. We also classify which of these CSPs can be expressed in Datalog. Our method relies on Ramsey theory; we prove that RCC5 has a Ramsey order expansion.

Killing a Vortex

Killing a Vortex

Classifying subset feedback vertex set for H-free graphs

In the Feedback Vertex Set problem, we aim to find a small set S of vertices in a graph intersecting every cycle. The Subset Feedback Vertex Set problem requires S to intersect only those cycles that include a vertex of some specified set T. We also consider the Weighted Subset Feedback Vertex Set problem, where each vertex u has weight w(u)>0 and we ask that S has small weight. By combining known NP-hardness results with new polynomial-time results we prove full complexity dichotomies for Subset Feedback Vertex Set and Weighted Subset Feedback Vertex Set for H-free graphs, that is, graphs that do not contain a graph H as an induced subgraph.

The Complexity of Finding Fair Many-to-One Matchings

We analyze the (parameterized) computational complexity of “fair” variants of bipartite many-to-one matching, where each vertex from the “left” side is matched to exactly one vertex and each vertex from the “right” side may be matched to multiple vertices. We want to find a “fair” matching, in which each vertex from the right side is matched to a “fair” set of vertices. Assuming that each vertex from the left side has one color modeling its “attribute”, we study two fairness criteria. For instance, in one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is, for instance, relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively. We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. We establish the correctness of our integer programs, based on Frank’s separation theorem [Frank, Discrete Math. 1982]. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.

Identity Complexity’s Influence on Multicultural Families’ Ethnic Identity Development and Acculturation Outcomes: A Qualitative Study among Binational (Estonian–Foreign) Parents in Estonia

For multicultural family members who live in cosmopolitan environments, concepts such as ethnic identity and integration have different significance. Some individuals can report, for example, that ethnic identity and integration have never played an important role in their lives and even feel that they represent old-fashioned notions from which modern societies should rather move on. For others, these concepts are much more relevant and are experienced in more challenging and complex ways. This article explores the influence that identity complexity—a cognitive disposition to perceive overlaps between different social identities, plays in this process. Forty parents of Estonian–foreign children (a traditionally cosmopolitan segment) were interviewed in Estonia and prompted to talk about topics such as their own ethnic identity(ies), their (and their family’s) feelings of integration into the Estonian society, and the way in which they represent their children’s ethnic identities, e.g., mostly Estonian/foreign, fifty–fifty, global citizen, etc. Thematic analysis combined with intersectionality suggests that there are associations between the identity complexity of interviewees and their attitudes towards these topics. Furthermore, results show that beyond the traditional dichotomy of high vs. low identity complexity, some interviewed parents have transitioned from higher to lower levels of identity complexity and vice versa at different times in their lives for different reasons. This study sheds light on identity complexity as a relevant predictor of acculturation and ethnic identity development outcomes among multicultural family members. It also contributes to the literature on cosmopolitan populations as a diverse group.

Complexity of the (Connected) Cluster Vertex Deletion Problem on H-free Graphs

The well-known Cluster Vertex Deletion problem (cluster-vd) asks for a given graph G and an integer k whether it is possible to delete a set S of at most k vertices of G such that the resulting graph G-S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G-S$$\\end{document} is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs H for which cluster-vd on H-free graphs is polynomially solvable and for which it is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-complete. Moreover, in the NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-completeness cases, cluster-vd cannot be solved in sub-exponential time in the vertex number of the H-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of cluster-vd, the Connected Cluster Vertex Deletion problem (connected cluster-vd), in which the set S has to induce a connected subgraph of G. It turns out that connected cluster-vd admits the same complexity dichotomy for H-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on H-free graphs.

“The Last Tram Has Gone”:

Jibanananda Das remains one of the major post-Tagore literary personas whose works are still open to interpretation. Torn between nurturing literary aestheticism and the responsibility to be a bread-earner, the complex dichotomy of Das’ life has shaped his writings significantly. The fictional writings also guide towards the complex labyrinth Das himself is. Like a mysterious kaleidoscope, he reflects light in the unexplored regions, and his writings bear the paradox a modern man faces when he is homeless. Likewise, according to György Lukács, the philosophical term “transcendental homelessness” expresses the yearning for a soulful and emotional home that is no longer available in this world. Nonetheless, Das’ oeuvre carries a fervent longing to be at home everywhere, and the yearning to find roots in a time of restlessness has left a permanent mark on his personal life as well. The identity crisis and alienation from society have forced him to continuously search for belongingness in a world full of fragmentation. Das’ alienation and innate desire to belong somewhere in Bengal portray the finest example of transcendental homelessness, which is evident in his collection of poems and stories. This research aims to study the inherent urge of Das to be at home everywhere and the sense of belongingness illustrated in his poems and stories.

A Complete Complexity Dichotomy of the Edge-Coloring Problem for All Sets of $$8$$-Edge Forbidden Subgraphs

A Complete Complexity Dichotomy of the Edge-Coloring Problem for All Sets of $$8$$-Edge Forbidden Subgraphs

(1,1)-Cluster Editing is polynomial-time solvable

A graph H is a clique graph if H is a vertex-disjoin union of cliques. Abu-Khzam (2017) introduced the (a,d)-Cluster Editing problem, where for fixed natural numbers a,d, given a graph G and vertex-weights a∗:V(G)→{0,1,…,a} and d∗:V(G)→{0,1,…,d}, we are to decide whether G can be turned into a cluster graph by deleting at most d∗(v) edges incident to every v∈V(G) and adding at most a∗(v) edges incident to every v∈V(G). Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of (a,d)-Cluster Editing for all pairs a,d apart from a=d=1. Abu-Khzam (2017) conjectured that (1,1)-Cluster Editing is in P. We resolve Abu-Khzam’s conjecture in affirmative by (i) providing a series of five polynomial-time reductions to C3-free and C4-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving (1,1)-Cluster Editing on C3-free and C4-free graphs of maximum degree at most 3.

Complexity classification of the eight-vertex model

We prove a complexity dichotomy theorem for the eight-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit. For tractability, we find some new classes of problems computable in polynomial time. For #P-hardness, we employ Möbius transformations to prove the success of interpolations.

The Complexity of the Matroid Homomorphism Problem

We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.

Public goods games in directed networks

Public goods games in undirected networks are generally known to have pure Nash equilibria, which are easy to find. In contrast, we prove that, in directed networks, a broad range of public goods games have intractable equilibrium problems: The existence of pure Nash equilibria is NP-hard to decide, and mixed Nash equilibria are PPAD-hard to find. We define general utility public goods games, and prove a complexity dichotomy result for finding pure equilibria, and a PPAD-completeness proof for mixed Nash equilibria. Even in the divisible goods variant of the problem, where existence is easy to prove, finding the equilibrium is PPAD-complete. Finally, when the treewidth of the directed network is appropriately bounded, we prove that polynomial-time algorithms are possible.

Bipartite 3-regular counting problems with mixed signs

We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. These are also counting CSP problems where every constraint has arity 3 and every variable is read-thrice. For every problem of the form Holant(f|=3), where f is any integer (or equivalently, rational)-valued ternary symmetric constraint function on Boolean variables, we prove that it is either P-time computable or #P-hard, depending on an explicit criterion of f. The constraint function can take both positive and negative values, allowing for cancellations. In addition, we discover a new phenomenon: there is a set F with the property that for every f∈F the problem Holant(f|=3) is planar P-time computable but #P-hard in general, yet its planar tractability is by a combination of a holographic transformation by [111−1] to FKT together with an independent global argument.

Dichotomy result on 3-regular bipartite non-negative functions

We prove a complexity dichotomy theorem for a class of Holant problems on 3-regular bipartite graphs. Given an arbitrary nonnegative weighted symmetric constraint function f=[x0,x1,x2,x3], we prove that the bipartite Holant problem Holant(f|(=3)) is either computable in polynomial time or #P-hard. The dichotomy criterion on f is explicit.

The complexity of blocking (semi)total dominating sets with edge contractions

We consider the problem of reducing the (semi)total domination number of a graph by one by contracting edges. It is known that this can always be done with at most three edge contractions and that deciding whether one edge contraction suffices is an NP-hard problem. We show that for every fixed k∈{2,3}, deciding whether exactly k edge contractions are necessary is NP-hard and further provide for k=2 complete complexity dichotomies on monogenic graph classes.

Decremental Optimization of Vertex-Coloring Under the Reconfiguration Framework

Suppose that we are given a positive integer k, and a k-(vertex-)coloring of a given graph G. Then we are asked to find a coloring of G using the minimum number of colors among colorings that are reachable from by iteratively changing a color assignment of exactly one vertex while maintaining the property of k-colorings. In this paper, we give linear-time algorithms to solve the problem for graphs of degeneracy at most two and for the case where . These results imply linear-time algorithms for series-parallel graphs and grid graphs. In addition, we give linear-time algorithms for chordal graphs and cographs. On the other hand, we show that, for any , this problem remains NP-hard for planar graphs with degeneracy three and maximum degree four. Thus, we obtain a complexity dichotomy for this problem with respect to the degeneracy of a graph and the number k of colors.