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

Many-to-one 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 = (AUP,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-toone 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-toone 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 classic polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP-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 umax as basis, and we prove that this dependence is necessary unless FPT = W[1]. Finally, we also present an approximation algorithm for the general case with performance guarantee umax+1, which is asymptotically best possible unless P = NP

Complexity and Approximability of Parameterized MAX-CSPs

International audienceWe study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance.We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a (1−ϵ)-approximation in time f(k,ϵ)⋅poly(n). 3. There is no FPTAS unless FPT=W[1].For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results

Parameterized Orientable Deletion

A graph is d-orientable if its edges can be oriented so that the maximum in-degree of the resulting digraph is at most d. d-orientability is a well-studied concept with close connections to fundamental graph-theoretic notions and applications as a load balancing problem. In this paper we consider the d-Orientable Deletion problem: given a graph G=(V,E), delete the minimum number of vertices to make G d-orientable. We contribute a number of results that improve the state of the art on this problem. Specifically: - We show that the problem is W[2]-hard and log n-inapproximable with respect to k, the number of deleted vertices. This closes the gap in the problem\u27s approximability. - We completely characterize the parameterized complexity of the problem on chordal graphs: it is FPT parameterized by d+k, but W-hard for each of the parameters d,k separately. - We show that, under the SETH, for all d,epsilon, the problem does not admit a (d+2-epsilon)^{tw}, algorithm where tw is the graph\u27s treewidth, resolving as a special case an open problem on the complexity of PseudoForest Deletion. - We show that the problem is W-hard parameterized by the input graph\u27s clique-width. Complementing this, we provide an algorithm running in time d^{O(d * cw)}, showing that the problem is FPT by d+cw, and improving the previously best know algorithm for this case

Parameterized Approximation Schemes using Graph Widths

Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or clique-width. Our main contribution is to present a natural randomized rounding technique that extends well-known ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, evading both polynomial-time inapproximability and parameterized intractability bounds