Related papers: Tight bounds and conjectures for the isolation lem…
The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its…
We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in (Mulmuley et al. 1987) achieves the…
In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, a function $\lambda: \mathcal V\rightarrow \mathbb…
A classic conjecture of F\"{u}redi, Kahn and Seymour (1993) states that given any hypergraph with non-negative edge weights $w(e)$, there exists a matching $M$ such that $\sum_{e \in M} (|e|-1+1/|e|)\, w(e) \geq w^*$, where $w^*$ is the…
Let H be a hypergraph on n vertices with the property that no edge contains another. We prove some results for a special case of the Isolation Lemma when the label set for the edges of H can only take two values. Given any set of vertices S…
The perfect matching problem has a randomized NC algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph, ensures that it…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
Let G be an edge-weighted hypergraph on n vertices, m edges of size \le s, where the edges have real weights in an interval [1,W]. We show that if we can approximate a maximum weight matching in G within factor alpha in time T(n,m,W) then…
An isolating set in a graph is a set $X$ of vertices such that every edge of the graph is incident with a vertex of $X$ or its neighborhood. The isolation number of a graph, or equivalently the vertex-edge domination number, is the minimum…
Given a fixed hypergraph $H$, let $\mbox{wsat}(n,H)$ denote the smallest number of edges in an $n$-vertex hypergraph $G$, with the property that one can sequentially add the edges missing from $G$, so that whenever an edge is added, a new…
Let $\alpha'$ and $\mu_i$ denote the matching number of a non-empty simple graph $G$ with $n$ vertices and the $i$-th smallest eigenvalue of its Laplacian matrix, respectively. In this paper, we prove a tight lower bound $$\alpha' \ge…
In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum…
A well-known conjecture of F\"uredi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights $w$, there exists a matching $M$ such that the inequality $\sum_{e\in M} g(e) w(e) \geq…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph $H$ and $\varepsilon > 0$, if an $n$-vertex graph $G$ contains $\varepsilon n^2$ edge-disjoint copies of $H$ then $G$ contains…
Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The…
Makespan minimization in restricted assignment $(R|p_{ij}\in \{p_j, \infty\}|C_{\max})$ is a classical problem in the field of machine scheduling. In a landmark paper in 1990 [8], Lenstra, Shmoys, and Tardos gave a 2-approximation algorithm…
Motivated by the exact weight perfect matching problem and recent parameterized algorithms for finding an $\ell$-th smallest perfect matching, we study structural properties of edge-weight symmetries in graphs. Recent work by El Maalouly et…
For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal value of various problems. However, there has not been as much success in…
Let $G = (V,E)$ be a graph, and for each $e \in E(G)$, let $L_e$ be a list of real numbers. Let $w:E(G) \to \cup_{e \in E(G)}L_e$ be an edge weighting function such that $w(e) \in L_e$ for each $e \in E(G)$, and let $c_w$ be the vertex…