arXiv subjects

S. Jukna

Publications and source records attributed to S. Jukna.

3 recordsLinked to original sources

arXiv2012

Clique problem, cutting plane proofs and communication complexity

Motivated by its relation to the length of cutting plane proofs for the Maximum Biclique problem, we consider the following communication game on a given graph G, known to both players. Let K be the maximal number of vertices in a complete bipartite subgraph of G, which is not necessarily an induced subgraph if G is not bipartite. Alice gets a set A of vertices, and Bob gets a disjoint set B of vertices such that |A|+|B|>K. The goal is to find a nonedge of G between A and B. We show that O(\log n) bits of communication are enough for every n-vertex graph.

cs.CC↗

arXiv2010

Min-Rank Conjecture for Log-Depth Circuits

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate of which can only depend on variables corresponding to *-entries in the i-th row of A. We conjecture that no such system can have more than 2^{n-c\cdot mr(A)} solutions, where c>0 is an absolute constant and mr(A) is the smallest rank over GF(2) of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x --> Mx. The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.

cs.CC↗

arXiv2010

Circuits with arbitrary gates for random operators

We consider boolean circuits computing n-operators f:{0,1}^n --> {0,1}^n. As gates we allow arbitrary boolean functions; neither fanin nor fanout of gates is restricted. An operator is linear if it computes n linear forms, that is, computes a matrix-vector product y=Ax over GF(2). We prove the existence of n-operators requiring about n^2 wires in any circuit, and linear n-operators requiring about n^2/\log n wires in depth-2 circuits, if either all output gates or all gates on the middle layer are linear.

cs.CC↗