Matchings with lower quotas: Algorithms and complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G=(A∪˙P,E)G=(A∪˙P,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NPNP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umaxumax as basis, and we prove that this dependence is necessary unless FPT=W[1]FPT=W[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee umax+1umax+1, which is asymptotically best possible unless P=NPP=NP. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas

Schulze and Ranked-Pairs Voting are Fixed-Parameter Tractable to Bribe, Manipulate, and Control

Schulze and ranked-pairs elections have received much attention recently, and the former has quickly become a quite widely used election system. For many cases these systems have been proven resistant to bribery, control, or manipulation, with ranked pairs being particularly praised for being NP-hard for all three of those. Nonetheless, the present paper shows that with respect to the number of candidates, Schulze and ranked-pairs elections are fixed-parameter tractable to bribe, control, and manipulate: we obtain uniform, polynomial-time algorithms whose degree does not depend on the number of candidates. We also provide such algorithms for some weighted variants of these problems

Cluster Editing Parameterized Above Modification-Disjoint P?-Packings

Given a graph G = (V,E) and an integer k, the Cluster Editing problem asks whether we can transform G into a union of vertex-disjoint cliques by at most k modifications (edge deletions or insertions). In this paper, we study the following variant of Cluster Editing. We are given a graph G = (V,E), a packing ? of modification-disjoint induced P?s (no pair of P?s in H share an edge or non-edge) and an integer ?. The task is to decide whether G can be transformed into a union of vertex-disjoint cliques by at most ?+|H| modifications (edge deletions or insertions). We show that this problem is NP-hard even when ? = 0 (in which case the problem asks to turn G into a disjoint union of cliques by performing exactly one edge deletion or insertion per element of H) and when each vertex is in at most 23 P?s of the packing. This answers negatively a question of van Bevern, Froese, and Komusiewicz (CSR 2016, ToCS 2018), repeated by C. Komusiewicz at Shonan meeting no. 144 in March 2019. We then initiate the study to find the largest integer c such that the problem remains tractable when restricting to packings such that each vertex is in at most c packed P?s. Van Bevern et al. showed that the case c = 1 is fixed-parameter tractable with respect to ? and we show that the case c = 2 is solvable in |V|^{2? + O(1)} time

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)

Anti-Factor Is FPT Parameterized by Treewidth and List Size (But Counting Is Hard)

In the general AntiFactor problem, a graph G and, for every vertex v of G, a set X_v ? ? of forbidden degrees is given. The task is to find a set S of edges such that the degree of v in S is not in the set X_v. Standard techniques (dynamic programming plus fast convolution) can be used to show that if M is the largest forbidden degree, then the problem can be solved in time (M+2)^{tw}?n^{O(1)} if a tree decomposition of width tw is given. However, significantly faster algorithms are possible if the sets X_v are sparse: our main algorithmic result shows that if every vertex has at most x forbidden degrees (we call this special case AntiFactor_x), then the problem can be solved in time (x+1)^{O(tw)}?n^{O(1)}. That is, AntiFactor_x is fixed-parameter tractable parameterized by treewidth tw and the maximum number x of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor? is already #W[1]-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set X, we denote by X-AntiFactor the special case where every vertex v has the same set X_v = X of forbidden degrees. We show the following lower bound for every fixed set X: if there is an ? > 0 such that #X-AntiFactor can be solved in time (max X+2-?)^{tw}?n^{O(1)} given a tree decomposition of width tw, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions