Weighted Random Popular Matchings

For a set A of n applicants and a set I of m items, we consider a problem of computing a matching of applicants to items, i.e., a function M mapping A to I; here we assume that each applicant $x \in A$ provides a preference list on items in I. We say that an applicant $x \in A$ prefers an item p than an item q if p is located at a higher position than q in its preference list, and we say that x prefers a matching M over a matching M' if x prefers M(x) over M'(x). For a given matching problem A, I, and preference lists, we say that M is more popular than M' if the number of applicants preferring M over M' is larger than that of applicants preferring M' over M, and M is called a popular matching if there is no other matching that is more popular than M. Here we consider the situation that A is partitioned into $A_{1},A_{2},...,A_{k}$, and that each $A_{i}$ is assigned a weight $w_{i}>0$ such that w_{1}>w_{2}>...>w_{k}>0$. For such a matching problem, we say that M is more popular than M' if the total weight of applicants preferring M over M' is larger than that of applicants preferring M' over M, and we call M an k-weighted popular matching if there is no other matching that is more popular than M. In this paper, we analyze the 2-weighted matching problem, and we show that (lower bound) if$m/n^{4/3}=o(1)$, then a random instance of the 2-weighted matching problem with$w_{1} \geq 2w_{2}$has a 2-weighted popular matching with probability o(1); and (upper bound) if$n^{4/3}/m = o(1)$, then a random instance of the 2-weighted matching problem with$w_{1} \geq 2w_{2}$ has a 2-weighted popular matching with probability 1-o(1).Comment: 13 pages, 2 figure

Improved Approximation Lower Bounds for TSP with Distances One and Two

The metric travelling salesman problem Δ-Tsp is the traveling salesman problem in which the distances among cities satisfy the triangle inequality. In this paper, we consider the matric traveling salesman problem Δ(1,2)-Tsp with distances one and two and Δ(1,2,3)-Tsp with distances one, two, and three as the special cases of Δ-Tsp. Since Δ(1,2)-Tsp is NP-complete, it is NP-hard to find an optimal solution for Δ(1,2)-Tsp. So in polynomial time, we with to find an approximate solution for Δ(1,2)-Tsp. owever Δ(1,2)-Tsp is APX-complete, there is a nontrivial approximation lower bound for Δ(1,2)-Tsp. For any ε>0, Engebretsen showed that it is NP-hard to approximate the symmetric Δ(1,2)-Tsp within 5381/5380-ε; the asymmetric Δ(1,2)-Tsp within 2805/2804-ε, and Bochenhauer, et al. showed that it is NP-hard to approximate the symmetric Δ(1,2,3)-Tsp within 3813/3812-ε. In this paper, we improve those lower bounds and show that for any ε>0, it is NP-hard to approximate the symmetric Δ(1,2)-Tsp within 1027/1026-ε (Corollary 4.5); the asymmetric Δ(1,2)-Tsp within 535/534-ε (Corollary 4.7); the symmetric Δ(1,2,3)-Tsp within 817/816-ε (Theorem 5.2); the asymmetric Δ(1,2,3)-Tsp within 364/363-ε (Theorem 5.3). We also show that for any ε>0, it is NP-hard to approximate Shortest-Superstring within 1279/1278-ε (Corollary 6.3)

New Constructions for Query-Efficient Locally Decodable Codes of Subexponential Length

A $(k,\delta,\epsilon)$-locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$ is an error-correcting code that encodes each message $\vec{x}=(x_{1},x_{2},...,x_{n}) \in F_{q}^{n}$ to $C(\vec{x}) \in F_{q}^{N}$ and has the following property: For any $\vec{y} \in {\bf F}_{q}^{N}$ such that $d(\vec{y},C(\vec{x})) \leq \delta N$ and each $1 \leq i \leq n$, the symbol $x_{i}$ of $\vec{x}$ can be recovered with probability at least $1-\epsilon$ by a randomized decoding algorithm looking only at $k$ coordinates of $\vec{y}$. The efficiency of a $(k,\delta,\epsilon)$-locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$ is measured by the code length $N$ and the number $k$ of queries. For any $k$-query locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$, the code length $N$ is conjectured to be exponential of $n$, however, this was disproved. Yekhanin [In Proc. of STOC, 2007] showed that there exists a 3-query locally decodable code $C: F_{2}^{n} \to F_{2}^{N}$ such that $N=\exp(n^{(1/\log \log n)})$ assuming that the number of Mersenne primes is infinite. For a 3-query locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$, Efremenko [ECCC Report No.69, 2008] reduced the code length further to $N=\exp(n^{O((\log \log n/ \log n)^{1/2})})$, and also showed that for any integer $r>1$, there exists a $k$-query locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$ such that $k \leq 2^{r}$ and $N=\exp(n^{O((\log \log n/ \log n)^{1-1/r})})$. In this paper, we present a query-efficient locally decodable code and show that for any integer $r>1$, there exists a $k$-query locally decodable code $C: F_{q}^{n} \to F_{q}^{N}$ such that $k \leq 3 \cdot 2^{r-2}$ and $N=\exp(n^{O((\log \log n/ \log n)^{1-1/r})})$.Comment: 13 pages, 1 figure, 2 table

Physical Zero-Knowledge Proof for Numberlink

Numberlink is a logic puzzle for which the player has to connect all pairs of cells with the same numbers by non-crossing paths in a rectangular grid. In this paper, we propose a physical protocol of zero-knowledge proof for Numberlink using a deck of cards, which allows a player to physically show that he/she knows a solution without revealing it. In particular, we develop a physical protocol to count the number of elements in a list that are equal to a given secret value without revealing that value, the positions of elements in the list that are equal to it, or the value of any other element in the list. Our protocol can also be applied to verify the existence of vertex-disjoint paths connecting all given pairs of endpoints in any undirected graph

Greedy Algorithms for Multi-Queue Buffer Management with Class Segregation

In this paper, we focus on a multi-queue buffer management in which packets of different values are segregated in different queues. Our model consists of m packets values and m queues. Recently, Al-Bawani and Souza (arXiv:1103.6049v2 [cs.DS] 30 Mar 2011) presented an online multi-queue buffer management algorithm Greedy and showed that it is 2-competitive for the general m-valued case, i.e., m packet values are 0 < v_{1} < v_{2} < ... < v_{m}, and (1+v_{1}/v_{2})-competitive for the two-valued case, i.e., two packet values are 0 < v_{1} < v_{2}. For the general m-valued case, let c_i = (v_{i} + \sum_{j=1}^{i-1} 2^{j-1} v_{i-j})/(v_{i+1} + \sum_{j=1}^{i-1}2^{j-1}v_{i-j}) for 1 \leq i \leq m-1, and let c_{m}^{*} = \max_{i} c_{i}. In this paper, we precisely analyze the competitive ratio of Greedy for the general m-valued case, and show that the algorithm Greedy is (1+c_{m}^{*})-competitive.Comment: 19 page