The Manickam-Miklós-Singhi Conjecture states that when n ≥ 4k, every multiset of n real numbers with nonnegative total sum has at least ( n−1 k−1 ) k-subsets with nonnegative sum. We develop a branch-and-cut strategy using a linear programming formulation to show that verifying the conjecture for fixed values of k is a finite problem. To improve our search, we develop a zero-error randomized pr...

For any integer k ? 2, we prove combinatorially the following Euler (binomial) transformation identity $${\rm{NC}}_{n + 1}^{(k)}(t) = t\sum\limits_{i 0}^n {\left({\matrix{n \cr i \cr}} \right)} {\rm{NW}}_i^{(k)}(t),$$ where NC () (t) (resp. NW (t)) is sum of weights, tnumber blocks, partitions {1,…,m} without k-crossings enhanced k-crossings). The special 2 and t 1 case, asserting Motzkin numbe...

For a graph G and a collection of vertex pairs {(s1, t1), . . . , (sk, tk)}, the k disjoint paths problem is to find k vertex-disjoint paths P1, . . . , Pk, where Pi is a path from si to ti for each i = 1, . . . , k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths Pi have to be chosen such that a given objective function is minimized. We...

Example 1. Show that every amount of change n ≥ 8 can be paid with only 3c and 5c coins. Proof: Suppose, for the sake of contradiction, that there are some values of n ≥ 8 such that it is not possible to pay n with 3c and 5c coins. Take the set of all such values. Since all of them are natural numbers, there is, by well-ordering principle, a minimal element in this set; let’s call it k. Now, co...

We present new algorithms for exact multilinear k-monomial counting which is to compute the sum of coefficients of all degree-k multilinear monomials in a given polynomial P over a ring R described by an arithmetic circuit C. If the polynomial can be represented as a product of two polynomials with degree at most d < k, our algorithm can solve this problem in O∗( ( n ↓d ) ) time, where ( n ↓d )...

We consider two closely related fundamental clustering problems in this paper. In the min-sum k-clustering one is given a metric space and has to partition the points into k clusterswhile minimizing the sum of pairwise distances between the points within the clusters. In theBalanced k-Median problem the instance is the same and one has to obtain a clustering into kcluster C1, . ...

1. WRIGHT* has shown that, if A^..., A, are positive numbers whose sum is 1 and if k ^ 3, a > (k— 2)2*~ + 5) then every sufficiently largen can be expressed as a sum n = wif+...+mJ of a positive kth powers such that \n-mt = 0(n^) (i = 1,2,..., a) with 0 < fi < a where a is given as a certain function of k and a; for example, if k = 3 and a = 9, then a = ^. In this paper we prove the following s...

This paper studies the sum rate performance of a low complexity quantized CSI-based Tomlinson-Harashima (TH) precoding scheme for downlink multiuser MIMO tansmission, employing greedy user selection. The asymptotic distribution of the output signal to interference plus noise ratio of each selected user and the asymptotic sum rate as the number of users K grows large are derived by using extreme...

The Equal Sum Subsets problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard. In this paper we give (pseudo-)polynomial algorithms and/or (strong) NP-hardness proofs for several natural variations of Equal Sum Subsets. Among others we present (1) a framework for obtaining NP-ha...

ion of shortest-paths problem: Definition 19. A commutative semi-ring is a set K and two operations ⊗,⊕ satisfying: 1. ⊕ is commutative and associative; there is an identity “0” s.t. k + 0 = k for all k ∈ K. 2. ⊗ is commutative and associative; there is a multiplicative identity “1” s.t. k ⊗ 1 = k for all k ∈ K. 3. The distributive law holds: a⊗ (b⊕ c) = (a⊗ b)⊕ (a⊗ c). Definition 20. The min-s...