arXiv subjects

Daniël Paulusma

Publications and source records attributed to Daniël Paulusma.

At least 37 records · Page 2Linked to original sources

arXiv2024

Partitioned Matching Games for International Kidney Exchange

We introduce partitioned matching games as a suitable model for international kidney exchange programmes, where in each round the total number of available kidney transplants needs to be distributed amongst the participating countries in a "fair" way. A partitioned matching game $(N,v)$ is defined on a graph $G=(V,E)$ with an edge weighting $w$ and a partition $V=V_1 \cup \dots \cup V_n$. The player set is $N = \{1, \dots, n\}$, and player $p \in N$ owns the vertices in $V_p$. The value $v(S)$ of a coalition $S \subseteq N$ is the maximum weight of a matching in the subgraph of $G$ induced by the vertices owned by the players in $S$. If $|V_p|=1$ for all $p\in N$, then we obtain the classical matching game. Let $c=\max\{|V_p| \; |\; 1\leq p\leq n\}$ be the width of $(N,v)$. We prove that checking core non-emptiness is polynomial-time solvable if $c\leq 2$ but co-NP-hard if $c\leq 3$. We do this via pinpointing a relationship with the known class of $b$-matching games and completing the complexity classification on testing core non-emptiness for $b$-matching games. With respect to our application, we prove a number of complexity results on choosing, out of possibly many optimal solutions, one that leads to a kidney transplant distribution that is as close as possible to some prescribed fair distribution.

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arXiv2023

Maximizing Matching Cuts

A matching cut in a graph G is an edge cut of G that is also a matching. This short survey gives an overview of old and new results and open problems for Maximum Matching Cut, which is to determine the size of a largest matching cut in a graph. We also compare this problem with the related problems Matching Cut, Minimum Matching Cut, and Perfect Matching Cut, which are to determine if a graph has a matching cut; the size of a smallest matching cut in a graph; and if a graph has a matching cut that is a perfect matching, respectively. Moreover, we discuss a relationship between Maximum Matching Cut and Max Cut, which is to determine the size of a largest edge cut in a graph, as well as a relationship between Minimum Matching Cut and Min Cut, which is to determine the size of a smallest edge cut in a graph.

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arXiv2023

Computing pivot-minors

A graph $G$ contains a graph $H$ as a pivot-minor if $H$ can be obtained from $G$ by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. Pivot-minors have mainly been studied from a structural perspective. In this paper we perform the first systematic computational complexity study of pivot-minors. We first prove that the Pivot-Minor problem, which asks if a given graph $G$ contains a pivot-minor isomorphic to a given graph $H$, is NP-complete. If $H$ is not part of the input, we denote the problem by $H$-Pivot-Minor. We give a certifying polynomial-time algorithm for $H$-Pivot-Minor when (1) $H$ is an induced subgraph of $P_3+tP_1$ for some integer $t\geq 0$, (2) $H=K_{1,t}$ for some integer $t\geq 1$, or (3) $|V(H)|\leq 4$ except when $H \in \{K_4,C_3+ P_1\}$. Let ${\cal F}_H$ be the set of induced-subgraph-minimal graphs that contain a pivot-minor isomorphic to $H$. To prove the above statement, we either show that there is an integer $c_H$ such that all graphs in ${\cal F}_H$ have at most $c_H$ vertices, or we determine ${\cal F}_H$ precisely, for each of the above cases.

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arXiv2023

Complexity Framework For Forbidden Subgraphs I: The Framework

For any particular class of graphs, algorithms for computational problems restricted to the class often rely on structural properties that depend on the specific problem at hand. This begs the question if a large set of such results can be explained by some common problem conditions. We propose such conditions for $HH$-subgraph-free graphs. For a set of graphs $HH$, a graph $G$ is $HH$-subgraph-free if $G$ does not contain any of graph from $H$ as a subgraph. Our conditions are easy to state. A graph problem must be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness must be preserved under edge subdivision of subcubic graphs. Our meta-classification says that if a graph problem satisfies all three conditions, then for every finite set $HH$, it is ``efficiently solvable'' on $HH$-subgraph-free graphs if $HH$ contains a disjoint union of one or more paths and subdivided claws, and is ``computationally hard'' otherwise. We illustrate the broad applicability of our meta-classification by obtaining a dichotomy between polynomial-time solvability and NP-completeness for many well-known partitioning, covering and packing problems, network design problems and width parameter problems. For other problems, we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). The proposed framework thus gives a simple pathway to determine the complexity of graph problems on $HH$-subgraph-free graphs. This is confirmed even more by the fact that along the way, we uncover and resolve several open questions from the literature.

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arXiv2023

The Complexity of Matching Games: A Survey

Matching games naturally generalize assignment games, a well-known class of cooperative games. Interest in matching games has grown recently due to some breakthrough results and new applications. This state-of-the-art survey provides an overview of matching games and extensions, such as $b$-matching games and partitioned matching games; the latter originating from the emerging area of international kidney exchange. In this survey we focus on computational complexity aspects of various game-theoretical solution concepts, such as the core, nucleolus and Shapley value, when the input is restricted to a matching game or one if its variants.

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arXiv2023

Complexity Framework for Forbidden Subgraphs III: When Problems are Tractable on Subcubic Graphs

For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity is determined on classes of $\mathcal{H}$-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most~$3$ and examine their complexity on $H$-subgraph-free graph classes where $H$ is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree $3$. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

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arXiv2023

Computing Balanced Solutions for Large International Kidney Exchange Schemes

To overcome incompatibility issues, kidney patients may swap their donors. In international kidney exchange programmes (IKEPs), countries merge their national patient-donor pools. We consider a recently introduced credit system. In each round, countries are given an initial ``fair'' allocation of the total number of kidney transplants. This allocation is adjusted by a credit function yielding a target allocation. The goal is to find a solution that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool. As solutions, we use maximum matchings that lexicographically minimize the country deviations from the target allocation. We perform, for the first time, a computational study for a {\it large} number of countries. For the initial allocations we use two easy-to-compute solution concepts, the benefit value and the contribution value, and four classical but hard-to-compute concepts, the Shapley value, nucleolus, Banzhaf value and tau value. By using state-of-the-art software we show that the latter four concepts are now within reach for IKEPs of up to fifteen countries. Our experiments show that using lexicographically minimal maximum matchings instead of ones that only minimize the largest deviation from the target allocation (as previously done) may make an IKEP up to 54% more balanced.

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arXiv2022

Solving Problems on Generalized Convex Graphs via Mim-Width

A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of $H$. Many $\mathsf{NP}$-complete problems, including problems such as Dominating Set, Feedback Vertex Set, Induced Matching and List $k$-Colouring, become polynomial-time solvable for ${\mathcal H}$-convex graphs when ${\mathcal H}$ is the set of paths. In this case, the class of ${\mathcal H}$-convex graphs is known as the class of convex graphs. The underlying reason is that the class of convex graphs has bounded mim-width. We extend the latter result to families of ${\mathcal H}$-convex graphs where (i) ${\mathcal H}$ is the set of cycles, or (ii) ${\mathcal H}$ is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least $3$. As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of ${\mathcal H}$-convex graphs is unbounded if ${\mathcal H}$ is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least $3$. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of ${\cal H}$-convex graphs.

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arXiv2022

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.

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arXiv2022

Induced Disjoint Paths and Connected Subgraphs for $H$-Free Graphs

Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.

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arXiv2022

On The Complexity of Matching Cut for Graphs of Bounded Radius and $H$-Free Graphs

For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most $d$ if $d\leq 2$ and NP-complete if $d\geq 3$. We prove the same dichotomy for graphs of bounded radius. For a graph $H$, a graph is $H$-free if it does not contain $H$ as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for $P_6$-free graphs, extending a recent result of Feghali for $P_5$-free graphs. We then extend our result to hold even for $(sP_3+P_6)$-free graphs for every $s\geq 0$ and initiate a complexity classification of Matching Cut for $H$-free graphs.

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arXiv2022

Finding Matching Cuts in $H$-Free Graphs

The NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on $H$-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant $r>0$ such that the first variant is NP-complete for $P_r$-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for $H$-free graphs.

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arXiv2022

Few Induced Disjoint Paths for $H$-Free Graphs

Paths $P^1,\ldots,P^k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P^i$ and $P^j$ have neither common vertices nor adjacent vertices. For a fixed integer $k$, the $k$-Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P^i$ such that each $P^i$ starts from $s_i$ and ends at $t_i$. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer $k$, a classical result from the literature states that even $2$-Induced Disjoint Paths is NP-complete. We prove new complexity results for $k$-Induced Disjoint Paths if the input is restricted to $H$-free graphs, that is, graphs without a fixed graph $H$ as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where $k$ is part of the input.

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arXiv2022

Partitioning H-Free Graphs of Bounded Diameter

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to $H$-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

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arXiv2022

An Algorithmic Framework for Locally Constrained Homomorphisms

A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.

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arXiv2022

Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. In particular, we prove that for every $s\geq 1$, both problems are polynomial-time solvable for $sP_3$-free graphs and $(sP_1+P_5)$-free graphs; here, the graph $sP_3$ denotes the disjoint union of $s$ paths on three vertices and the graph $sP_1+P_5$ denotes the disjoint union of $s$ isolated vertices and a path on five vertices. Our new results for Feedback Vertex Set extend all known polynomial-time results for Feedback Vertex Set on $H$-free graphs, namely for $sP_2$-free graphs [Chiarelli et al., TCS 2018], $(sP_1+P_3)$-free graphs [Dabrowski et al., Algorithmica 2020] and $P_5$-free graphs [Abrishami et al., SODA 2021]. Together, the new results also show that both problems exhibit the same behaviour on $H$-free graphs (subject to some open cases). This is in part due to a new general algorithm we design for finding in a ($sP_3)$-free or $(sP_1+P_5)$-free graph $G$ a largest induced subgraph whose blocks belong to some finite class ${\cal C}$ of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on $H$-free graphs.

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arXiv2021

Computing Subset Transversals in $H$-Free Graphs

We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to $H$-free graphs, that is, to graphs that do not contain some fixed graph~$H$ as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on $H$-free graphs for every graph $H$ except when $H=sP_1+P_4$ for some $s\geq 1$. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for $(sP_1+P_4)$-free graphs for every $s\geq 1$.

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arXiv2021

Induced Disjoint Paths in AT-free Graphs

Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ connects $s_i$ and $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We study it for AT-free graphs. Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for AT-free graphs, we prove that it can be solved in polynomial time for AT-free graphs even when $k$ is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs, for which such a result only holds if $k$ is fixed. As a consequence of our main result, the problem of deciding if a given AT-free graph contains a fixed graph $H$ as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter $|V_H|$, even on a subclass of AT-free graph, namely cobipartite graphs. We also show that the problems $k$-in-a-Path and $k$-in-a-Tree are polynomial-time solvable on AT-free graphs even if $k$ is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning $k$ given vertices.

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