اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدMatching-Vector Families and LDCs Over Large Modulo
54
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove new upper bounds on the size of families of vectors in $Z_m^n$ with restricted modular inner products, when $m$ is a large integer. More formally, if $vec{u}_1,ldots,vec{u}_t in Z_m^n$ and $vec{v}_1,ldots,vec{v}_t in Z_m^n$ satisfy $langlevec{u}_i,vec{v}_irangleequiv0pmod m$ and $langlevec{u}_i,vec{v}_jrangle otequiv0pmod m$ for all $i eq jin[t]$, we prove that $t leq O(m^{n/2+8.47})$. This improves a recent bound of $t leq m^{n/2 + O(log(m))}$ by cite{BDL13} and is the best possible up to the constant 8.47 when $m$ is sufficiently larger than $n$. The maximal size of such families, called `Matching-Vector families, shows up in recent constructions of locally decodable error correcting codes (LDCs) and determines the rate of the code. Using our result we are able to show that these codes, called Matching-Vector codes, must have encoding length at least $K^{19/18}$ for $K$-bit messages, regardless of their query complexity. This improves a known super linear bound of $ K2^{Omega({sqrt{log K}})}$ proved in cite{DGY11}.
قيم البحث
اقرأ أيضاً
A path in an(a) edge(vertex)-colored graph is called emph{a conflict-free path} if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called emph{conflict-free (vertex-)connected} if there is a conflict-
free path between each pair of distinct vertices. We call the graph $G$ emph{strongly conflict-free connected }if there exists a conflict-free path of length $d_G(u,v)$ for every two vertices $u,vin V(G)$. And the emph{strong conflict-free connection number} of a connected graph $G$, denoted by $scfc(G)$, is defined as the smallest number of colors that are required to make $G$ strongly conflict-free connected. In this paper, we first investigate the question: Given a connected graph $G$ and a coloring $c: E(or V)rightarrow {1,2,cdots,k} (kgeq 1)$ of the graph, determine whether or not $G$ is, respectively, conflict-free connected, vertex-conflict-free connected, strongly conflict-free connected under coloring $c$. We solve this question by providing polynomial-time algorithms. We then show that it is NP-complete to decide whether there is a k-edge-coloring $(kgeq 2)$ of $G$ such that all pairs $(u,v)in P (Psubset Vtimes V)$ are strongly conflict-free connected. Finally, we prove that the problem of deciding whether $scfc(G)leq k$ $(kgeq 2)$ for a given graph $G$ is NP-complete.
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains
at least $n!$ matrices of size $n times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA 21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollobas, and Morris [Eur. J. Comb. 06] on the growth of hereditary classes of ordered graphs. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width.
The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the Disjoint P
aths problem for $H$-free graphs. If $k$ is fixed, we obtain the $k$-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every $k geq 1$. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of $k$-Disjoint Connected Subgraphs for $H$-free graphs, and give the same almost-complete classification for Disjoint Connected Subgraphs for $H$-free graphs as for Disjoint Paths.
We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of G, in determining if there exists a sequence S=<D_1:=D_s,...,D_l:=D_t> of dominating sets of G
such that for any two consecutive dominating sets D_r and D_{r+1} with r<t, D_{r+1}=(D_r u) U v, where uv is an edge of G. In a recent paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that the problem is polynomial on circular-arc graphs and PSPACE-complete on circle graphs.
This report documents the program and the outcomes of Dagstuhl Seminar 13082 Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices, held in February 2013 at Dagstuhl Castle.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
