Polynomial-time Solvable Cases Research Articles

On conflict-free cuts: Algorithms and complexity

One way to define the Matching Cut problem is: Given a graph G, is there an edge-cut M of G such that M is an independent set in the line graph of G? We propose the more general Conflict-Free Cut problem: Together with the graph G, we are given a so-called conflict graph Gˆ on the edges of G, and we ask for an edge-cutset M of G that is independent in Gˆ. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is NP-complete even when the maximum degree of G is 5 and Gˆ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of G as a parameter, and we show W[1]-hardness even when G has a feedback vertex set of size one, and the clique cover number of Gˆ is the parameter. Since the clique cover number of Gˆ is an upper bound on the independence number of Gˆ and thus the solution size, this implies W[1]-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT.

An exact borderline between the NP-hard and polynomial-time solvable cases of flow shop scheduling with job-dependent storage requirements

An exact borderline between the NP-hard and polynomial-time solvable cases of flow shop scheduling with job-dependent storage requirements

The price-elastic knapsack problem

This paper introduces the price-elastic knapsack problem (PEKP), an extension of the classic knapsack problem where instead of fixed item characteristics, the weight of each item and the profit from including an item in the knapsack are a function of a parameter, namely the price. PEKP is first formulated as a generic nonlinear optimization problem and three special cases are investigated. First, we show a polynomial-time solvable case. Next, we formulate the case where the item weights are affine-linear functions as a quadratic program. The computational results show that solving the quadratic program to optimality is computationally challenging, and thus, an approach that decomposes the problem into three mixed integer programs is proposed. Similarly, the case where the weights of the items are piecewise-linear functions is investigated and a quadratic formulation is proposed. A solution approach based on decomposing the problem into three mixed integer programs that are solved independently is also proposed. Using randomly generated instances of varying sizes, the computational results show that the proposed decomposition leads to significant computational advantages compared to solving the quadratic program in the cases of affine-linear and piecewise-linear functions.

On Recognizing Staircase Compatibility

For the problem to find an m-clique in an m-partite graph, staircase compatibility has recently been introduced as a polynomial-time solvable special case. It is a property of a graph together with an m-partition of the vertex set and total orders on each subset of the partition. In optimization problems involving m-cliques in m-partite graphs as a subproblem, it allows for totally unimodular linear programming formulations, which have shown to efficiently solve problems from different applications. In this work, we address questions concerning the recognizability of this property in the case where the m-partition of the graph is given, but suitable total orders are to be determined. While finding these total orders is NP-hard in general, we give several conditions under which it can be done in polynomial time. For bipartite graphs, we present a polynomial-time algorithm to recognize staircase compatibility and show that staircase total orders are unique up to a small set of reordering operations. On m-partite graphs, where the recognition problem is NP-complete in the general case, we identify a polynomially solvable subcase and also provide a corresponding algorithm to compute the total orders. Finally, we evaluate the performance of our ordering algorithm for m-partite graphs on a set of artificial instances as well as real-world instances from a railway timetabling application. It turns out that applying the ordering algorithm to the real-world instances and subsequently solving the problem via the aforementioned totally unimodular reformulations indeed outperforms a generic formulation which does not exploit staircase compatibility.

On a borderline between the NP-hard and polynomial-time solvable cases of the flow shop with job-dependent storage requirements

The paper is concerned with the two-machine flow shop, where each job requires an additional resource (referred to as storage space) from the start of its first operation till the end of its second operation. The storage requirement of a job is determined by the processing time of its first operation. At any point in time, the total consumption of this additional resource cannot exceed a given limit (referred to as the storage capacity). The goal is to minimise the makespan, i.e. to minimise the time needed for the completion of all jobs. This problem is NP-hard in the strong sense. The paper analyses how the parameter - a lower bound on the storage capacity specified in terms of the processing times, affects the computational complexity.

Bribery and Control in Stable Marriage

We initiate the study of external manipulations in Stable Marriage by considering several manipulative actions as well as several manipulation goals. For instance, one goal is to make sure that a given pair of agents is matched in a stable solution, and this may be achieved by the manipulative action of reordering some agents' preference lists. We present a comprehensive study of the computational complexity of all problems arising in this way. We find several polynomial-time solvable cases as well as NP-hard ones. For the NP-hard cases, focusing on the natural parameter "budget" (that is, the number of manipulative actions one is allowed to perform), we also conduct a parameterized complexity analysis and encounter mostly parameterized hardness results.

Finding Diverse Trees, Paths, and More

Mathematical modeling is a standard approach to solve many real-world problems and diversity of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies have been devoted to finding diverse solutions. Baste et al. (Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse solutions of combinatorial problems from the perspective of fixed-parameter tractability. They considered problems of finding r solutions that maximize some diversity measures (the minimum or sum of the pairwise Hamming distances among them) and gave some fixed-parameter tractable algorithms for the diverse version of several well-known problems, such as Vertex Cover, Feedback Vertex Set, d-Hitting Set}, and problems on bounded-treewidth graphs. In this work, we further investigate the (fixed-parameter) tractability of problems of finding diverse spanning trees, paths, and several subgraphs. In particular, we show that, given a graph G and an integer r, the problem of computing r spanning trees of G maximizing the sum of the pairwise Hamming distances among them can be solved in polynomial time. To the best of the authors' knowledge, this is the first polynomial-time solvable case for finding diverse solutions of unbounded size.

Minimum Reload Cost Graph Factors

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to the problem of finding an r-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of r = 2, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of r and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT-algorithm.

Comparing temporal graphs using dynamic time warping

Within many real-world networks, the links between pairs of nodes change over time. Thus, there has been a recent boom in studying temporal graphs. Recognizing patterns in temporal graphs requires a proximity measure to compare different temporal graphs. To this end, we propose to study dynamic time warping on temporal graphs. We define the dynamic temporal graph warping (dtgw) distance to determine the dissimilarity of two temporal graphs. Our novel measure is flexible and can be applied in various application domains. We show that computing the dtgw-distance is a challenging (in general) NP-hard optimization problem and identify some polynomial-time solvable special cases. Moreover, we develop a quadratic programming formulation and an efficient heuristic. In experiments on real-world data, we show that the heuristic performs very well and that our dtgw-distance performs favorably in de-anonymizing networks compared to other approaches.

Stable Matchings with Covering Constraints: A Complete Computational Trichotomy

Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are mathsf {NP}-hard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440–465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixed-parameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal to 1, which generalizes the setting of “arranged marriages” first considered by Knuth (Mariages stables et leurs relations avec d’autres problèmes combinatoires, Les Presses de l’Université de Montréal, Montreal, 1976). We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy: for each choice of parameters we either provide a polynomial-time algorithm, or an mathsf {NP}-hardness proof and fixed-parameter algorithm, or mathsf {NP}-hardness proof and mathsf {W}[1]-hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440–465, 2016) by showing fixed-parameter intractability parameterized by optimal solution size. We also classify all cases of one-sided constraints where only women may be distinguished.

Efficient algorithms for ridesharing of personal vehicles

Given a set of trips in a road network, where each trip has an individual, a vehicle and some requirements, the ridesharing problem is to select a subset of vehicles to deliver the individuals of all trips to their destinations satisfying the requirements. Common requirements of a trip include the source, destination, vehicle capacity, detour distance limit, preferred paths, and time constraints of the trip. Minimizing the total travel distance of the vehicles and minimizing the number of vehicles are major optimization goals. These minimization problems are complex and NP-hard because each trip may have many requirements. We study simplified minimization problems in which each trip's requirements are specified by the source, destination, vehicle capacity, detour distance and preferred path parameters. We show that both minimization problems can be solved in polynomial time if all of the following conditions are satisfied: (1) all trips have the same destination or same source; (2) no detour is allowed and (3) each trip has one unique preferred path. It is known that both minimization problems are NP-hard if any one of the three conditions is not satisfied. Our results and the NP-hard results suggest a boundary between the polynomial time solvable cases and NP-hard cases for the minimization problems.

Complexity and relaxation methods for minimising total average cycle stock subject to practical constraints

This paper considers the problem of minimising total average cycle stock that is subject to practical constraints, as first studied by Silver and Moon and later by Hsieh, and Billionnet. For the problem, reorder intervals of a population of items are restricted to a given set, and the total number of replenishments allowed per unit time is limited. Previous researchers proposed different mathematical programming formulations and relaxation methods without identifying the computational complexity of the problem. In this study, we investigate the computational complexity of the problem and analyse the proposed relaxation methods. We identify NP-hard and polynomial time solvable cases of the problem and compare three different relaxations in terms of the lower bounds provided by each relaxation method. We also show that the relaxation with the strongest bound can be solved using a linear time greedy algorithm instead of a general-purpose linear programming algorithm.

Settlement fund circulation problem

In the economic activities, the central bank has an important role to cover payments of banks, when they are short of funds to clear their debts. For this purpose, the central bank timely puts funds so that the economic activities go smooth. Since payments in this mechanism are processed sequentially, the total amount of funds put by the central bank critically depends on the order of the payments. Then an interest goes to the amount to prepare if the order of the payments can be controlled by the central bank, or if it is determined under the worst case scenario. This motivates us to introduce a brand-new problem, which we call the settlement fund circulation problem. The problems are formulated as follows: Let G=(V,A) be a directed multigraph with a vertex set V and an arc set A. Each arc a∈A is endowed debt d(a)≥0, and the debts are settled sequentially under a sequence π of arcs. Each vertex v∈V is put fund in the amount of pπ(v)≥0 under the sequence. The minimum/maximum settlement fund circulation problem (Min-SFC/Max-SFC) in a given graph G with debts d:A→R+∪{0} asks to find a bijection π:A→{1,2,…,|A|} that minimizes/maximizes the total funds ∑v∈Vpπ(v). In this paper, we show that both Min-SFC and Max-SFC are NP-hard; in particular, Min-SFC is (I) strongly NP-hard even if G is (i) a multigraph with |V|=2 or (ii) a simple graph with treewidth at most two, and is (II) (not necessarily strongly) NP-hard for simple trees of diameter four, while it is solvable in polynomial time for stars. Also, we identify several polynomial time solvable cases for both problems.

Reconfiguration of colorable sets in classes of perfect graphs

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (called a reconfiguration sequence) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

The quadratic M-convexity testing problem

M-convex functions, which are a generalization of valuated matroids, play a central role in discrete convex analysis. Quadratic M-convex functions constitute a basic and important subclass of M-convex functions, which has a close relationship with phylogenetics as well as valued constraint satisfaction problems. In this paper, we consider the quadraticM-convexity testing problem (QMCTP), which is the problem of deciding whether a given quadratic function on {0,1}n is M-convex. We show that QMCTP is co-NP-complete in general, but is polynomial-time solvable under a natural assumption. Furthermore, we propose an O(n2)-time algorithm for solving QMCTP in the polynomial-time solvable case.

Clustering with Lower-Bounded Sizes

Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios, however, the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for $$k=2$$k=2 with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k.

Algorithmic analysis for ridesharing of personal vehicles

The ridesharing problem is to share personal vehicles by individuals with similar itineraries. A trip in the ridesharing problem is an individual and his/her itinerary. To realize a trip is to deliver the individual to his/her destination by a vehicle satisfying the itinerary requirement. Major optimization goals in the ridesharing problem include minimizing the number of vehicles and minimizing the total travel distance of vehicles to realize all trips. There are many parameters in the minimization problems, making them complex and NP-hard. The problems can be simplified by considering only some of the parameters. We give an algorithmic analysis for the simplified minimization problems and explore a boundary between the NP-hard and polynomial time solvable cases. We prove that the simplified minimization problems, where only the source, destination, vehicle capacity, detour distance and preferred path parameters are considered, are still NP-hard. We show that the simplified problem of minimizing the number of vehicles becomes polynomial time solvable if the considered parameters satisfy certain conditions. These suggest a boundary between the NP-hard and polynomial time solvable cases.

Flow shop scheduling problem with conflict graphs

This paper addresses the problem of scheduling jobs on flow shops subject to constraints given by an undirected graph, in which each edge joins a pair of conflicting jobs that cannot be processed simultaneously on different machines. We first show that the problem of minimizing the maximum completion time (makespan) is NP-hard for several versions. Then, we present polynomial-time solvable cases. On the other hand, we propose heuristic approaches and lower bounds alongside with an experimental study.

Open shop scheduling problems with conflict graphs

This paper deals with open shops subject to the constraint that some conflicting jobs cannot be processed simultaneously on different machines. In the context of our study, these conflicts are given by an undirected graph, called the conflict graph. We seek a schedule that minimizes the maximum completion time (makespan). We first prove the NP-hardness of different versions of this problem. Then, we present polynomial-time solvable cases, for which we devise simple and efficient algorithms. On the other hand, we present heuristics and lower bounds for the general case. The heuristics are based on the construction of matchings in bipartite graphs as well as on a specific insertion technique combined with beam search. Finally, we test the efficiency of the heuristics and report our computational experiments that display satisfying results.

Computational complexity of distance edge labeling

The problem of Distance Edge Labeling is a variant of Distance Vertex Labeling (also known as L2,1 labeling) that has been studied for more than twenty years and has many applications, such as frequency assignment.The Distance Edge Labeling problem asks whether the edges of a given graph can be labeled such that the labels of adjacent edges differ by at least two and the labels of edges at distance two differ by at least one. Labels are chosen from the set {0,1,…,λ} for λ fixed.We present a full classification of its computational complexity—a dichotomy between the polynomial-time solvable cases and the remaining cases which are NP-complete. We characterize graphs with λ≤4 which leads to a polynomial-time algorithm recognizing the class and we show NP-completeness for λ≥5 by several reductions from Monotone Not All Equal 3-SAT.Moreover, there is an absolute constant c>0 such that there is no 2cn-time algorithm deciding the Distance Edge Labeling problem unless the exponential time hypothesis fails.