127,726 research outputs found
Hardness of submodular cost allocation : lattice matching and a simplex coloring conjecture
We consider the Minimum Submodular Cost Allocation (MSCA) problem. In this problem, we are given k submodular cost functions f1, ... , fk: 2V -> R+ and the goal is to partition V into k sets A1, ..., Ak so as to minimize the total cost sumi = 1,k fi(Ai). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function fi satisfies fi = gi + h, where each gi is monotone submodular and h is (possibly non-monotone) submodular. We give an O(k log |V|) approximation for this problem. We provide some evidence that a factor of k may be necessary, even in the special case of HyperLabel. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon) for k-uniform HyperLabel and label set [k]. We provide a proof of the simplex-coloring conjecture for k=3
Efficient algorithms for reconfiguration in VLSI/WSI arrays
The issue of developing efficient algorithms for reconfiguring processor arrays in the presence of faulty processors and fixed hardware resources is discussed. The models discussed consist of a set of identical processors embedded in a flexible interconnection structure that is configured in the form of a rectangular grid. An array grid model based on single-track switches is considered. An efficient polynomial time algorithm is proposed for determining feasible reconfigurations for an array with a given distribution of faulty processors. In the process, it is shown that the set of conditions in the reconfigurability theorem is not necessary. A polynomial time algorithm is developed for finding feasible reconfigurations in an augmented single-track model and in array grid models with multiple-track switche
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=(AâȘËP,E)G=(AâȘËP,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
X THEN X: Manipulation of Same-System Runoff Elections
Do runoff elections, using the same voting rule as the initial election but
just on the winning candidates, increase or decrease the complexity of
manipulation? Does allowing revoting in the runoff increase or decrease the
complexity relative to just having a runoff without revoting? For both weighted
and unweighted voting, we show that even for election systems with simple
winner problems the complexity of manipulation, manipulation with runoffs, and
manipulation with revoting runoffs are independent, in the abstract. On the
other hand, for some important, well-known election systems we determine what
holds for each of these cases. For no such systems do we find runoffs lowering
complexity, and for some we find that runoffs raise complexity. Ours is the
first paper to show that for natural, unweighted election systems, runoffs can
increase the manipulation complexity
An O(n log n)-Time Algorithm for the Restricted Scaffold Assignment
The assignment problem takes as input two finite point sets S and T and
establishes a correspondence between points in S and points in T, such that
each point in S maps to exactly one point in T, and each point in T maps to at
least one point in S. In this paper we show that this problem has an O(n log
n)-time solution, provided that the points in S and T are restricted to lie on
a line (linear time, if S and T are presorted).Comment: 13 pages, 8 figure
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