Related papers: Is the Algorithmic Kadison-Singer Problem Hard?
We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called $k$-LCIS: Given $k$ integer sequences $X_1,\dots,X_k$ of length at most $n$, the task is to determine the…
In this paper we suggest analytical methods and associated algorithms for determining the sum of the subsets $X_m$ of the set $X_n$ (subset sum problem). Our algorithm has time complexity $T=O(C_{n}^{k})$ ($k=[m/2]$, which significantly…
Longest Common Substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decisional version of this problem, LCS with threshold $d$, asks whether two length-$n$ input…
The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by…
We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give…
The Shortest Common Superstring problem (SCS) consists, for a set of strings S = {s_1,...,s_n}, in finding a minimum length string that contains all s_i, 1<= i <= n, as substrings. While a 2+11/30 approximation ratio algorithm has recently…
Motivated by computing duplication patterns in sequences, a new fundamental problem called the longest subsequence-repeated subsequence (LSRS) is proposed. Given a sequence $S$ of length $n$, a letter-repeated subsequence is a subsequence…
Determinantal Point Processes (DPPs) are probabilistic models that arise in quantum physics and random matrix theory and have recently found numerous applications in computer science. DPPs define distributions over subsets of a given ground…
In this work, we consider a variant of the classical Longest Common Subsequence problem called Doubly-Constrained Longest Common Subsequence (DC-LCS). Given two strings s1 and s2 over an alphabet A, a set C_s of strings, and a function Co…
In this paper, we seek a natural problem and a natural distribution of instances such that any $O(n^{c-\epsilon})$-time algorithm fails to solve most instances drawn from the distribution, while the problem admits an $n^{c+o(1)}$-time…
We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of $m$ sets over $n$ elements in the query model, we show that sub-linear algorithms derived from existing techniques have…
In this paper, we focus on the problem of existence and computing of small and large stable models. We show that for every fixed integer k, there is a linear-time algorithm to decide the problem LSM (large stable models problem): does a…
We investigate parameterizing hard combinatorial problems by the size of the solution set compared to all solution candidates. Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in…
Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms…
We study the problem of strongly refuting semirandom $k$-LIN$(\mathbb{F})$ instances: systems of $k$-sparse inhomogeneous linear equations over a finite field $\mathbb{F}$. For the case of $\mathbb{F} = \mathbb{F}_2$, this is the…
The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$…
Unbounded SubsetSum is a classical textbook problem: given integers $w_1,w_2,\cdots,w_n\in [1,u],~c,u$, we need to find if there exists $m_1,m_2,\cdots,m_n\in \mathbb{N}$ satisfying $c=\sum_{i=1}^n w_im_i$. In its all-target version, $t\in…
Finding the common subsequences of $L$ multiple strings has many applications in the area of bioinformatics, computational linguistics, and information retrieval. A well-known result states that finding a Longest Common Subsequence (LCS)…
Simon's problem asks the following: determine if a function $f: \{0,1\}^n \rightarrow \{0,1\}^n$ is one-to-one or if there exists a unique $s \in \{0,1\}^n$ such that $f(x) = f(x \oplus s)$ for all $x \in \{0,1\}^n$, given the promise that…
We give a nontrivial algorithm for the satisfiability problem for cn-wire threshold circuits of depth two which is better than exhaustive search by a factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first nontrivial…