On the number of nearly perfect matchings in almost regular uniform hypergraphs

AbstractStrengthening the result of Rődl and Frankl (Europ. J. Combin 6 (1985) 317–326), Pippenger proved the theorem stating the existence of a nearly perfect matching in almost regular uniform hypergraph satisfying some conditions (see J. Combin. Theory A 51 (1989) 24–42). Grable announced in J. Combin. Designs 4 (4) (1996) 255–273 that such hypergraphs have exponentially many nearly perfect matchings. This generalizes the result and the proof in Combinatorica 11 (3) (1991) 207–218 which is based on the Rődl Nibble algorithm (European J. Combin. 5 (1985) 69–78). In this paper, we present a simple proof of Grable's extension of Pippenger's theorem. Our proof is based on a comparison of upper and lower bounds of the probability for a random subgraph to have a nearly perfect matching. We use the Lovasz Local Lemma to obtain the desired lower bound of this probability

Identically self-blocking clutters

A clutter is identically self-blocking if it is equal to its blocker. We prove that every identically self-blocking clutter different from is nonideal. Our proofs borrow tools from Gauge Duality and Quadratic Programming. Along the way we provide a new lower bound for the packing number of an arbitrary clutter

Combinatorial Auctions Do Need Modest Interaction

We study the necessity of interaction for obtaining efficient allocations in subadditive combinatorial auctions. This problem was originally introduced by Dobzinski, Nisan, and Oren (STOC'14) as the following simple market scenario: $m$ items are to be allocated among $n$ bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: in each round, each bidder, simultaneously with others, broadcasts a message to all parties involved and the central planner computes an allocation solely based on the communicated messages. Dobzinski et.al. showed that no non-interactive ($1$-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than $\Omega(m^{{1}/{4}})$, while for any $r \geq 1$, there exists $r$-round protocols that achieve $\widetilde{O}(r \cdot m^{{1}/{r+1}})$ approximation with polynomial communication; in particular, $O(\log{m})$ rounds of interaction suffice to obtain an (almost) efficient allocation. A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain an efficient allocation. In this paper, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: we show that for any $r \geq 1$, any $r$-round protocol that uses polynomial communication can only approximate the social welfare up to a factor of $\Omega(\frac{1}{r} \cdot m^{{1}/{2r+1}})$. This in particular implies that $\Omega(\frac{\log{m}}{\log\log{m}})$ rounds of interaction are necessary for obtaining any efficient allocation in these markets. Our work builds on the recent multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein (FOCS'15) and settles an open question posed by Dobzinski et.al. and Alon et. al

Probabilistic methods in combinatorial and stochastic optimization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (leaves 103-106).(cont.) Packing/Covering problems, we prove upper and lower bounds on the adaptivity gap depending on the dimension. We also design polynomial-time algorithms achieving near-optimal approximation guarantees with respect to the adaptive optimum. Finally, we prove complexity-theoretic results regarding optimal adaptive policies. These results are based on a connection between adaptive policies and Arthur-Merlin games which yields PSPACE-hardness results for numerous questions regarding adaptive policies.In this thesis we study a variety of combinatorial problems with inherent randomness. In the first part of the thesis, we study the possibility of covering the solutions of an optimization problem on random subgraphs. The motivation for this approach is a situation where an optimization problem needs to be solved repeatedly for random instances. Then we seek a pre-processing stage which would speed-up subsequent queries by finding a fixed sparse subgraph covering the solution for a random subgraph with high probability. The first problem that we investigate is the minimum spanning tree. Our essential result regarding this problem is that for every graph with edge weights, there is a set of O(n log n) edges which contains the minimum spanning tree of a random subgraph with high probability. More generally, we extend this result to matroids. Further, we consider optimization problems based on the shortest path metric and we find covering sets of size 0(n(Ì1+2/c) log2Ì n) that approximate the shortest path metric of a random vertex-induced subgraph within a constant factor of c with high probability. In the second part, we turn to a model of stochastic optimization, where a solution is built sequentially by selecting a collection of "items". We distinguish between adaptive and non-adaptive strategies, where adaptivity means being able to perceive the precise characteristics of chosen items and use this knowledge in subsequent decisions. The benefit of adaptivity is our central concept which we investigate for a variety of specific problems. For the Stochastic Knapsack problem, we prove constant upper and lower bounds on the "adaptivity gap" between optimal adaptive and non-adaptive policies. For more general Stochasticby Jan Vondrák.Ph.D

OPTIMA OF DUAL INTEGER LINEAR PROGRAMS

We consider dual pairs of packing and covering integer linear programs. Best possible bounds are found between their optimal values. Tight inequalities are obtained relating the integral optima and the optimal rational solutions