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Register a new userA Tale of Santa Claus, Hypergraphs and Matroids
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Sami Davies
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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A well-known problem in scheduling and approximation algorithms is the Santa Claus problem. Suppose that Santa Claus has a set of gifts, and he wants to distribute them among a set of children so that the least happy child is made as happy as possible. Here, the value that a child $i$ has for a present $j$ is of the form $p_{ij} in { 0,p_j}$. A polynomial time algorithm by Annamalai et al. gives a $12.33$-approximation and is based on a modification of Haxells hypergraph matching argument. In this paper, we introduce a matroid version of the Santa Claus problem. Our algorithm is also based on Haxells augmenting tree, but with the introduction of the matroid structure we solve a more general problem with cleaner methods. Our result can then be used as a blackbox to obtain a $(4+varepsilon)$-approximation for Santa Claus. This factor also compares against a natural, compact LP for Santa Claus.
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Perturbed graphic matroids are binary matroids that can be obtained from a graphic matroid by adding a noise of small rank. More precisely, r-rank perturbed graphic matroid M is a binary matroid that can be represented in the form I +P, where I is the incidence matrix of some graph and P is a binary matrix of rank at most r. Such matroids naturally appear in a number of theoretical and applied settings. The main motivation behind our work is an attempt to understand which parameterized algorithms for various problems on graphs could be lifted to perturbed graphic matroids. We study the parameterized complexity of a natural generalization (for matroids) of the following fundamental problems on graphs: Steiner Tree and Multiway Cut. In this generalization, called the Space Cover problem, we are given a binary matroid M with a ground set E, a set of terminals Tsubseteq E, and a non-negative integer k. The task is to decide whether T can be spanned by a subset of Esetminus T of size at most k. We prove that on graphic matroid perturbations, for every fixed r, Space Cover is fixed-parameter tractable parameterized by k. On the other hand, the problem becomes W[1]-hard when parameterized by r+k+|T| and it is NP-complete for rleq 2 and |T|leq 2. On cographic matroids, that are the duals of graphic matroids, Space Cover generalizes another fundamental and well-studied problem, namely Multiway Cut. We show that on the duals of perturbed graphic matroids the Space Cover problem is fixed-parameter tractable parameterized by r+k.
We address counting and optimization variants of multicriteria global min-cut and size-constrained min-$k$-cut in hypergraphs. 1. For an $r$-rank $n$-vertex hypergraph endowed with $t$ hyperedge-cost functions, we show that the number of multiobjective min-cuts is $O(r2^{tr}n^{3t-1})$. In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne (Math Programming, 2015). In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 2. We also address node-budgeted multiobjective min-cuts: For an $n$-vertex hypergraph endowed with $t$ vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is $O(r2^{r}n^{t+2})$, where $r$ is the rank of the hypergraph, and the number of node-budgeted $b$-multiobjective min-cuts for a fixed budget-vector $b$ is $O(n^2)$. 3. We show that min-$k$-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant $k$, thus resolving an open problem posed by Queyranne. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger (SODA, 1993). Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained $k$-cuts in hypergraphs.
We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. The input to the Weighted Diverse Bases problem consists of a matroid $M$, a weight function $omega:E(M)tomathbb{N}$, and integers $kgeq 1, dgeq 0$. The task is to decide if there is a collection of $k$ bases $B_{1}, dotsc, B_{k}$ of $M$ such that the weight of the symmetric difference of any pair of these bases is at least $d$. This is a diverse variant of the classical matroid base packing problem. The input to the Weighted Diverse Common Independent Sets problem consists of two matroids $M_{1},M_{2}$ defined on the same ground set $E$, a weight function $omega:Etomathbb{N}$, and integers $kgeq 1, dgeq 0$. The task is to decide if there is a collection of $k$ common independent sets $I_{1}, dotsc, I_{k}$ of $M_{1}$ and $M_{2}$ such that the weight of the symmetric difference of any pair of these sets is at least $d$. This is motivated by the classical weighted matroid intersection problem. The input to the Diverse Perfect Matchings problem consists of a graph $G$ and integers $kgeq 1, dgeq 0$. The task is to decide if $G$ contains $k$ perfect matchings $M_{1},dotsc,M_{k}$ such that the symmetric difference of any two of these matchings is at least $d$. We show that Weighted Diverse Bases and Weighted Diverse Common Independent Sets are both NP-hard, and derive fixed-parameter tractable (FPT) algorithms for all three problems with $(k,d)$ as the parameter.
Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $lceil frac{n}{2} rceil$, which is best possible. Our result settles a special case of a strengthening of the Caccetta-Haggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.
In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get rid of it, and if we choose not to accept it then we cannot later add it. Babaioff, Immorlica, and Kleinberg [SODA 2007] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)-competitive algorithm. However, most matroids that are known to admit an O(1)-competitive algorithm can be easily represented using graphs (e.g. graphic and transversal matroids). In particular, there is very little known about F-representable matroids (the class of matroids that can be represented as elements of a vector space over a field F), which are one of the foundational matroid classes. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving an O(1)-competitive algorithm for regular matroids (the class of matroids that are representable over every field), and use techniques that are matroid-theoretic rather than graph-theoretic. We use the regular matroid decomposition theorem of Seymour to decompose any regular matroid into matroids which are either graphic, cographic, or isomorphic to R_{10}, and then show how to combine algorithms for these basic classes into an algorithm for regular matroids. This allows us to generalize beyond regular matroids to any class of matroids that admits such a decomposition into classes for which we already have good algorithms. In particular, we give an O(1)-competitive algorithm for the class of max-flow min-cut matroids.
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