Towards More Practical Linear Programming-based Techniques for Algorithmic Mechanism Design

R. Lavy and C. Swamy (FOCS 2005, J. ACM 2011) introduced a general method for obtaining truthful-in-expectation mechanisms from linear programming based approximation algorithms. Due to the use of the Ellipsoid method, a direct implementation of the method is unlikely to be efficient in practice. We propose to use the much simpler and usually faster multiplicative weights update method instead. The simplification comes at the cost of slightly weaker approximation and truthfulness guarantees

On Randomized Fictitious Play for Approximating Saddle Points Over Convex Sets

Given two bounded convex sets X\subseteq\RR^m and Y\subseteq\RR^n, specified by membership oracles, and a continuous convex-concave function F:X\times Y\to\RR, we consider the problem of computing an \eps-approximate saddle point, that is, a pair $(x^*,y^*)\in X\times Y$ such that \sup_{y\in Y} F(x^*,y)\le \inf_{x\in X}F(x,y^*)+\eps. Grigoriadis and Khachiyan (1995) gave a simple randomized variant of fictitious play for computing an \eps-approximate saddle point for matrix games, that is, when $F$ is bilinear and the sets $X$ and $Y$ are simplices. In this paper, we extend their method to the general case. In particular, we show that, for functions of constant "width", an \eps-approximate saddle point can be computed using O^*(\frac{(n+m)}{\eps^2}\ln R) random samples from log-concave distributions over the convex sets $X$ and $Y$. It is assumed that $X$ and $Y$ have inscribed balls of radius $1/R$ and circumscribing balls of radius $R$. As a consequence, we obtain a simple randomized polynomial-time algorithm that computes such an approximation faster than known methods for problems with bounded width and when \eps \in (0,1) is a fixed, but arbitrarily small constant. Our main tool for achieving this result is the combination of the randomized fictitious play with the recently developed results on sampling from convex sets

Characterization of the Vertices and Extreme Directions of the Negative Cycles Polyhedron and Hardness of Generating Vertices of 0/1-Polyhedra

Given a graph $G=(V,E)$ and a weight function on the edges w:E\mapsto\RR, we consider the polyhedron $P(G,w)$ of negative-weight flows on $G$, and get a complete characterization of the vertices and extreme directions of $P(G,w)$. As a corollary, we show that, unless $P=NP$, there is no output polynomial-time algorithm to generate all the vertices of a 0/1-polyhedron. This strengthens the NP-hardness result of Khachiyan et al. (2006) for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes \cite{BL98} [Bussieck and L\"ubbecke (1998)]

Conflict-Free Coloring Made Stronger

In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can be Conflict-Free colored with a total of at most $O(\log n)$ colors. That is, it can be colored with $O(\log n)$ colors such that for any (covered) point $p$ there is some disc whose color is distinct from all other colors of discs containing $p$. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be colored with a total of at most $O(k \log n)$ colors such that (a) for any point $p$ that is covered by at least $k$ discs, there are at least $k$ distinct discs each of which is colored by a color distinct from all other discs containing $p$ and (b) for any point $p$ covered by at most $k$ discs, all discs covering $p$ are colored distinctively. We call such a coloring a {\em $k$-Strong Conflict-Free} coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. \item [(ii)] More generally, for families of $n$ simple closed Jordan regions with union-complexity bounded by $O(n^{1+\alpha})$, we prove that there exists a $k$-Strong Conflict-Free coloring with at most $O(k n^\alpha)$ colors. \item [(iii)] We prove that any set of $n$ axis-parallel rectangles can be $k$-Strong Conflict-Free colored with at most $O(k \log^2 n)$ colors. \item [(iv)] We provide a general framework for $k$-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of $k$-Strong Conflict-Free coloring and the recently studied notion of $k$-colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings

On Tree-Constrained Matchings and Generalizations

We consider the following \textsc{Tree-Constrained Bipartite Matching} problem: Given two rooted trees $T_1=(V_1,E_1)$, $T_2=(V_2,E_2)$ and a weight function $w: V_1\times V_2 \mapsto \mathbb{R}_+$, find a maximum weight matching $\mathcal{M}$ between nodes of the two trees, such that none of the matched nodes is an ancestor of another matched node in either of the trees. This generalization of the classical bipartite matching problem appears, for example, in the computational analysis of live cell video data. We show that the problem is $\mathcal{APX}$-hard and thus, unless $\mathcal{P} = \mathcal{NP}$, disprove a previous claim that it is solvable in polynomial time. Furthermore, we give a $2$-approximation algorithm based on a combination of the local ratio technique and a careful use of the structure of basic feasible solutions of a natural LP-relaxation, which we also show to have an integrality gap of $2-o(1)$. In the second part of the paper, we consider a natural generalization of the problem, where trees are replaced by partially ordered sets (posets). We show that the local ratio technique gives a $2k\rho$-approximation for the $k$-dimensional matching generalization of the problem, in which the maximum number of incomparable elements below (or above) any given element in each poset is bounded by $\rho$. We finally give an almost matching integrality gap example, and an inapproximability result showing that the dependence on $\rho$ is most likely unavoidable

Pengaruh Lama Penyimpanan dan Konsentrasi Filtrat Daun Lidah Buaya (Aloe vera) Terhadap Mutu Buah Tomat (Lycopersicum esculentum)

Pembusukan merupakan salah satu masalah yang ditimbulkan setelah masa panen. Buah tomat akan terus mengalami penurunan mutu selama proses penyimpanan, pencegahan dapat dilakukan dengan pemberian edible coating. Salah satu bahan yang dapat dijadikan edible coating adalah daun lidah buaya. Lidah buaya mengandung senyawa antrakuinon, saponin dan emodin yang berfungsi sebagai antimikroba, antikuman dan antibakteri. Penelitian ini bertujuan untuk mengetahui pengaruh lama penyimpanan dan konsentrasi filtrat daun lidah buaya terhadap susut bobot, intensitas kecacatan, kandungan vitamin C dan pH selama proses penyimpanan dan untuk mengetahui berapakah konsentrasi lidah buaya yang paling efektif sebagai bahan pengawet. Kegiatan penelitian dilakukan melalui True Experimental Research. Tempat dan waktu penelitian dilaksanakan di Laboratorium Kimia Universitas Muhammadiyah Malang yang berlangsung pada tanggal 2 – 11 Agustus 2016. Rancangan penelitian yang digunakan adalah Rancangan Acak Lengkap (RAL) Terdiri dari 2 faktor yaitu lama penyimpanan dan konsentrasi filtrat dengan 3 kali ulangan. Analisis data menggunakan SPSS versi 16 dengan analisis varians satu arah dan uji Duncan pada taraf signifikansi 0,05. Hasil penelitian menunjukkan ada pengaruh lama penyimpanan dan konsentrasi filtrat daun lidah buaya terhadap susut bobot, intensitas kecacatan, dan kandungan vitamin C selama proses penyimpanan. Konsentrasi terbaik untuk perubahan susut bobot, intensitas kecacatan dan kandungan vitamin C terdapat pada konsentrasi 8%. Sedangkan untuk nilai pH tidak memiliki perbedaan pada tiap konsentrasi. Hasil penelitian dikembangkan menjadi media audiovisual materi bioteknologi pelajaran biologi SMA kelas XII

An Algorithm for Dualization in Products of Lattices and Its Applications

Let \cL=\cL_1×⋅s×\cL_n be the product of n lattices, each of which has a bounded width. Given a subset \cA\subseteq\cL, we show that the problem of extending a given partial list of maximal independent elements of \cA in \cL can be solved in quasi-polynomial time. This result implies, in particular, that the problem of generating all minimal infrequent elements for a database with semi-lattice attributes, and the problem of generating all maximal boxes that contain at most a specified number of points from a given n-dimensional point set, can both be solved in incremental quasi-polynomial time