NP-completeness Results Research Articles

Hardness transitions and uniqueness of acyclic colouring

For k∈N, a k-acyclic colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that (i) f(u)≠f(v) for every edge uv of G, and (ii) there is no cycle in G bicoloured by f. For k∈N, the problem k-Acyclic Colourability takes a graph G as input and asks whether G admits a k-acyclic colouring. Ochem (EuroComb 2005) proved that 3-Acyclic Colourability is NP-complete for bipartite graphs of maximum degree 4. Mondal et al. (2013) proved that 4-Acyclic Colourability is NP-complete for graphs of maximum degree five. We prove that for k≥3, k-Acyclic Colourability is NP-complete for bipartite graphs of maximum degree k+1, thereby generalising the NP-completeness result of Ochem, and adding bipartiteness to the NP-completeness result of Mondal et al.. In contrast, k-Acyclic Colourability is polynomial-time solvable for graphs of maximum degree at most 0.38k3/4. Hence, for k≥3, the least integer d such that k-Acyclic Colourability in graphs of maximum degree d is NP-complete, denoted by La(k), satisfies 0.38k3/4<La(k)≤k+1. We prove that for k≥4, k-Acyclic Colourability in d-regular graphs is NP-complete if and only if La(k)≤d≤2k−3. We also show that it is coNP-hard to check whether an input graph G admits a unique k-acyclic colouring up to colour swaps (resp. up to colour swaps and automorphisms).

Transitivity on subclasses of bipartite graphs

Let \(G=(V, E)\) be a graph where V and E are the vertex and edge sets, respectively. For two disjoint subsets A and B, we say A dominates B if every vertex of B is adjacent to at least one vertex of A. A vertex partition \(\pi = \{V_1, V_2, \ldots , V_k\}\) of G is called a transitive k-partition if \(V_i\) dominates \(V_j\) for all i, j where \(1\le i<j\le k\). The maximum integer k for which the above partition exists is called transitivity of G and it is denoted by Tr(G). The Maximum Transitivity Problem is to find a transitive partition of a given graph with the maximum number of parts. It was known that the decision version of Maximum Transitivity Problem is NP-complete for chordal graphs, which was proved by Hedetniemi et al. (Discret Math 278:81–108, 2004)]. This paper first strengthens the NP-completeness result by showing that this problem remains NP-complete for perfect elimination bipartite graphs. On the other hand, we propose a linear-time algorithm for finding the transitivity of a given bipartite chain graph. We then characterize graphs with transitivity at least t for any integer t. This result answers two open questions posed by J. T. Hedetniemi and S. T. Hedetniemi (J Combin Math Combin Comput 104:75–91, 2018).

Four proofs of the directed Brooks' Theorem

We give four new proofs of the directed version of Brook's Theorem and an NP-completeness result.

Dominating induced matching in some subclasses of bipartite graphs

A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. An edge e∈E is said to dominate itself and all other edges adjacent to it. A matching M in a graph G=(V,E) is called a dominating induced matching (d.i.m.) if every edge of G is dominated by edges of M exactly once. The dominating induced matching decide (DIM-Decide) problem asks whether a graph G contains a dominating induced matching. The dominating induced matching (DIM) problem asks to compute a dominating induced matching (d.i.m.) in a graph G that admits a dominating induced matching. The DIM-Decide problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of the DIM-Decide problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. On the positive side, we characterize the class of star-convex bipartite graphs admitting a d.i.m. This characterization leads to a linear time algorithm to test whether a star-convex bipartite graph admits a d.i.m. and, if so, constructs a d.i.m. in such a star-convex bipartite graph in linear time. We also propose polynomial time algorithms to construct a d.i.m. in long-k-star-convex bipartite graphs as well as in circular-convex bipartite graphs if the input graph admits a d.i.m.

Two-machine open shop problem with a single server and set-up time considerations

We address in this paper the two-machine open shop problem with set-up time considerations to build schedules that minimize the overall completion time, known as the makespan. In the model we are considering, the set-up, handled by a single server, consists of a preparation phase separated from the processing phase. Once the server completes the preparation of a job, it becomes again available for setting up other jobs, leaving the current one to complete its processing. We prove several NP-completeness results, exhibit a linear time algorithm for a special case, and develop heuristic and hyper-heuristic algorithms to solve the general problem along with an experimental study.

Global total k-domination: Approximation and hardness results

A subset D⊆V of a graph G=(V,E) is called a global total k-dominating set of G if D is a total k-dominating set of both G and the complement G‾ of G. The Minimum Global Totalk-Domination problem is to find a global total k-dominating set of minimum cardinality of the input graph G and Decide Global Totalk-Domination problem is the decision version of Minimum Global Totalk-Domination problem. The Decide Global Totalk-Domination problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of Decide Global Totalk-Domination problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, star-convex bipartite graphs and doubly chordal graphs. On the positive side, we give a polynomial time algorithm for the Minimum Global Totalk-Domination problem for chordal bipartite graphs, which is a subclass of bipartite graphs. We propose a 2(1+ln⁡|V|)-approximation algorithm for the Minimum Global Totalk-Domination problem for any graph. We show that Minimum Global Totalk-Domination problem cannot be approximated within (1−ϵ)ln⁡|V| for any ϵ>0 unless P=NP for any integer k≥1. We further show that for bipartite graphs, Minimum Global Totalk-Domination problem cannot be approximated within (16−ϵ)ln⁡|V| for any ϵ>0 unless P=NP for any k≥1. Finally, we show that the Minimum Global Totalk-Domination problem is APX-complete for bounded degree bipartite graphs for any fixed integer k≥1.

Hedonic Diversity Games Revisited

A hedonic diversity game (HDG) is a coalition formation problem, where the set of agents is partitioned into two types of agents (say red and blue agents), and each agent has preferences over the relative number (fraction) of agents of her own type in her coalition. In a dichotomous hedonic diversity game (DHDG) each agent partitions the set of possible fractions into a set of approved and a set of disapproved fractions. The solution concepts for these games considered in the literature so far are concerned with stability notions such as core and Nash stability. We add to the existing literature by providing NP-completeness results for the decision problems whether a DHDG admits (i) a Nash stable outcome and (ii) a strictly core stable outcome respectively, in restricted settings with only two (and three, respectively) approved fractions per agent. In addition, we consider the criterion of social welfare maximization in HDG and DHDG. In that context we provide an NP-completeness result for HDG under the use of Borda scores. For DHDG and approval scores, we draw the separation line between polynomially solvable and NP-complete cases with respect to the number of approved fractions per agent.

Separation of cycle inequalities in periodic timetabling

Cycle inequalities play an important role in the polyhedral study of the periodic timetabling problem in public transport. We give the first pseudo-polynomial time separation algorithm for cycle inequalities, and we contribute a rigorous proof for the pseudo-polynomial time separability of the change-cycle inequalities. Moreover, we provide several NP-completeness results, indicating that pseudo-polynomial time is best possible. The efficiency of these cutting planes is demonstrated on real-world instances of the periodic timetabling problem.

Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

Deciding if a graph has a square root is a classical problem, which has been studied extensively both from graph-theoretic and algorithmic perspective. As the problem is NP-complete, substantial effort has been dedicated to determining the complexity of deciding if a graph has a square root belonging to some specific graph class $${{\mathcal {H}}}$$ . There are both polynomial-time solvable and NP-complete results in this direction, depending on $${{\mathcal {H}}}$$ . We present a general framework for the problem if $$\mathcal{H}$$ is a class of sparse graphs. This enables us to generalize a number of known results and to give polynomial-time algorithms for the cases where $${{\mathcal {H}}}$$ is the class of outerplanar graphs and $${{\mathcal {H}}}$$ is the class of graphs of pathwidth at most 2.

Computing NodeTrix Representations of Clustered Graphs

NodeTrix representations are a popular way to visualize clustered graphs; they represent clusters as adjacency matrices and inter-cluster edges as curves connecting the matrix boundaries. We study the complexity of constructing NodeTrix representations focusing on planarity testing problems, and we show several $\mathbb{NP}$-completeness results and some polynomial-time algorithms. Building on such algorithms we develop a JavaScript library for NodeTrix representations aimed at reducing the crossings between edges incident to the same matrix.

Restrained Domination in Some Subclasses of Chordal Graphs

A set D⊆V of a graph G = (V, E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V\\D. The Minimum Restrained Domination problem is to find a restrained dominating set of minimum cardinality. The decision version of the Minimum Restrained Domination problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this NP-completeness result by showing that the problem remains NP-complete for doubly chordal graphs, a subclass of chordal graphs. We also propose a polynomial time algorithm to solve the Minimum Restrained Domination problem in block graphs, a subclass of doubly chordal graphs.

Knapsack in Graph Groups

It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for every graph group. This result even holds if the group elements are represented in a compressed form by so called straight-line programs, which generalizes the classical NP-completeness result of the integer knapsack problem. If group elements are represented explicitly by words over the generators, then knapsack for a graph group belongs to the class LogCFL (a subclass of P) if the graph group can be built up from the trivial group using the operations of free product and direct product with $\mathbb {Z}$ . In all other cases, the knapsack problem is NP-complete.

Improved complexity results on [formula omitted]-coloring [formula omitted]-free graphs

A graph is H-free if it does not contain an induced subgraph isomorphic to H. We denote by Pt and Ct the path and the cycle on t vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. The second result is the first NP-hardness result for k-COLORING P6-free graphs. These two results improve all previous NP-complete results on k-coloring Pt-free graphs, and almost complete the classification of complexity of k-COLORING Pt-free graphs for k≥4 and t≥1, leaving as the only missing case 4-COLORING of P6-free graphs. We expect that 4-COLORING is polynomial time solvable for P6-free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING P6-free graphs which are also banner-free, where banner is the graph obtained from C4 by adding a new vertex and making it adjacent to exactly one vertex on the C4.

Complexity of total outer-connected domination problem in graphs

A set D⊆V is called a total dominating set of a graph G=(V,E), if for all v∈V, |NG(v)∩D|≥1. A total dominating set D of G=(V,E) is called a total outer-connected dominating set if G[V∖D] is connected. The Minimum Total Outer-connected Domination problem for a graph G=(V,E) is to find a total outer-connected dominating set of minimum cardinality. The Total Outer-connected Domination Decision problem, the decision version of the Minimum Total Outer-connected Domination problem, is known to be NP-complete for general graphs. In this paper, we strengthen this NP-completeness result by showing that the Total Outer-connected Domination Decision problem remains NP-complete for chordal graphs, split graphs, and doubly chordal graphs. We prove that the Total Outer-connected Domination Decision problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for computing the total outer-connected domination number, the cardinality of a minimum total outer-connected dominating set, of a tree. We prove that the Minimum Total Outer-connected Domination problem cannot be approximated within a factor of (1−ε)ln|V| for any ε>0, unless NP ⊆ DTIME (|V|O(loglog|V|)). Finally, we show that the Minimum Total Outer-connected Domination problem is APX-complete for bounded degree graphs.

Weighted Electoral Control

Although manipulation and bribery have been extensively studied under weighted voting, there has been almost no work done on election control under weighted voting. This is unfortunate, since weighted voting appears in many important natural settings. In this paper, we study the complexity of controlling the outcome of weighted elections through adding and deleting voters. We obtain polynomial-time algorithms, NP-completeness results, and for many NP-complete cases, approximation algorithms. In particular, for scoring rules we completely characterize the complexity of weighted voter control. Our work shows that for quite a few important cases, either polynomial-time exact algorithms or polynomial-time approximation algorithms exist.

Algorithmic aspects of open neighborhood location–domination in graphs

A set D⊆V of a graph G=(V,E) is called an open neighborhood locating–dominating set (OLD-set) if (i) NG(v)∩D≠0̸ for all v∈V, and (ii) NG(u)∩D≠NG(v)∩D for every pair of distinct vertices u,v∈V. Given a graph G=(V,E), the Min OLD-set problem is to find an OLD-set of minimum cardinality. Given a graph G=(V,E) and a positive integer k, the Decide OLD-set problem is to decide whether G has an OLD-set of cardinality at most k. The Decide OLD-set problem is known to NP-complete for general graphs. In this paper we extend the NP-completeness result of the Decide OLD-set problem by showing that it remains NP-complete for bipartite graphs, planar graphs, split graphs and doubly chordal graphs. We prove that the Decide OLD-set problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for the Min OLD-set problem in trees. We also propose a (2+3lnΔ)-approximation algorithm for the Min OLD-set problem and show that the Min OLD-set problem cannot be approximated within 12(1−ϵ)ln|V| for any ϵ>0 unless NP⊆DTIME(|V|O(loglog|V|)). Finally, we prove that the Min OLD-set problem is APX-complete for bipartite graphs of maximum degree 3.

On the complexity of [formula omitted]-domination and k-tuple domination in graphs

We consider two types of graph domination—{k}-domination and k-tuple domination, for a fixed positive integer k—and provide new NP-complete as well as polynomial time solvable instances for their related decision problems. Regarding NP-completeness results, we solve the complexity of the {k}-domination problem on split graphs, chordal bipartite graphs and planar graphs, left open in 2008. On the other hand, by exploiting Courcelle's results on Monadic Second Order Logic, we obtain that both problems are polynomial time solvable for graphs with clique-width bounded by a constant. In addition, we give an alternative proof for the linearity of these problems on strongly chordal graphs.

Influence of the tie-break rule on the end-vertex problem

End-vertices of a given graph search may have some nice properties, as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see Rose, Tarjan and Lueker 1976. Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in Corneil, Köhler and Lanlignel 2010. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NP-completeness result for Breadth First Search solving a problem raised in Corneil, Köhler and Lanlignel 2010.

On the complexity of synthesizing a minimum-weighted supervisor under partial observation

With full observation the problem of synthesizing a minimum-weighted supervisor has been shown polynomial-time solvable. With partial observation the problem becomes NP-hard. In this paper we will show that the decision version of the problem becomes NP-complete when the natural projection is a natural observer. This NP-completeness result is unexpected, considering that the logic supervisor synthesis problem under the same assumption becomes polynomial-time solvable.

Coordination of production and interstage batch delivery with outsourced distribution

In this paper, we consider coordinated production and interstage batch delivery scheduling problems, where a third-party logistics provider (3PP) delivers semi-finished products in batches from one production location to another production location belonging to the same manufacturer. A batch cannot be delivered until all jobs of the batch are completed at the upstream stage. The 3PP is required to deliver each product within a time T from its release at the upstream stage. We consider two transportation modes: regular transportation, for which delivery departure times are fixed at the beginning, and express transportation, for which delivery departure times are flexible. We analyze the problems faced by the 3PP when either the manufacturer dominates or the 3PP dominates. In this context, we investigate the complexity of several problems, providing polynomiality and NP-completeness results.