Hsiang Hsuan Liu

Scheduling for Electricity Cost in Smart Grid

We study an offline scheduling problem arising in demand response management in smart grid. Consumers send in power requests with a flexible set of timeslots during which their requests can be served. For example, a consumer may request the dishwasher to operate for one hour during the periods 8am to 11am or 2pm to 4pm. The grid controller, upon receiving power requests, schedules each request within the specified duration. The electricity cost is measured by a convex function of the load in each timeslot. The objective of the problem is to schedule all requests with the minimum total electricity cost. As a first attempt, we consider a special case in which the power requirement and the duration a request needs service are both unit-size. For this problem, we present a polynomial time offline algorithm that gives an optimal solution and show that the time complexity can be further improved if the given set of timeslots is a contiguous interval.

On independence domination

Let G be a graph. The independence-domination number γi(G) is the maximum over all independent sets I in G of the minimal number of vertices needed to dominate I. In this paper we investigate the computational complexity of γi(G) for graphs in several graph classes related to cographs. We present an exact exponential algorithm. We show that there is a polynomial-time algorithm to compute a maximum independent set in the Cartesian product of two cographs. We prove that independence domination is NP-hard for planar graphs and we present a PTAS.

Rainbow Domination and Related Problems on Some Classes of Perfect Graphs

Let \(k \in \mathbb {N}\) and let G be a graph. A function \(f: V(G) \rightarrow 2^{[k]}\) is a rainbow function if, for every vertex x with \(f(x)=\varnothing \), \(f(N(x)) =[k]\), where [k] denotes the integers ranging from 1 to k. The rainbow domination number \(\gamma _{kr}(G)\) is the minimum of \(\sum _{x \in V(G)} |f(x)|\) over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs.

On maximum independent set of categorical product and ultimate categorical ratios of graphs

We first present polynomial algorithms to compute the independence number of the categorical product for two cographs or two splitgraphs, respectively. Then we prove that computing the maximum independent set of the categorical product of a planar graph of maximum degree three and a K 4 is NP-complete. The ultimate categorical independence ratio of a graph G is defined as lim k ? ∞ ? α ( G k ) / n k . The ultimate categorical independence ratio can be computed in polynomial time for cographs, splitgraphs, permutation graphs, interval graphs and graphs of bounded treewidth. Also, we present an O * ( 3 n / 3 ) -time exact, exponential algorithm for the ultimate categorical independence ratio of general graphs. We further present a PTAS for the ultimate categorical independence ratio of planar graphs. Lastly, we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).

On the Grundy number of Cameron graphs

The Grundy number of a graph is the maximal number of colors attained by a first-fit coloring of the graph. The class of Cameron graphs is the Seidel switching class of cographs. In this paper we show that the Grundy number is computable in polynomial time for Cameron graphs.

Convex Independence in Permutation Graphs

A set C of vertices of a graph is \(P_3\)-convex if every vertex outside C has at most one neighbor in C. The convex hull \(\sigma (A)\) of a set A is the smallest \(P_3\)-convex set that contains A. A set M is convexly independent if for every vertex \(x \in M\), \(x \notin \sigma (M-x)\). We show that the maximal number of vertices that a convexly independent set in a permutation graph can have, can be computed in polynomial time. (Due to space limit, the missing proofs are presented in the full paper. Please see https://drive.google.com/file/d/0B1Ilu0-p1dDsSkpsZFZsR1Y4Uk0/view or http://arxiv.org/abs/1609.02657).

On the P 3 -convexity of some classes of graphs with few P 4 s and permutation graphs

We analyze the P3-geodetic number, the P3-hull number and the P3-Carathéodory number for tree-cographs, and these parameters for P4-reducible graphs. We also show that the P3-hull number is polynomial for permutation graphs. Moreover, we give monadic second-order formulas for the P3-hull and P3-Carathéodory numbers in general, thus showing that these parameters are polynomial for graphs of bounded rankwidth or treewidth.

On complexities of minus domination

A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of a graph G=(V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x \in V. The minus-domination number \gamma^{-}(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1. In this paper we show that the minus-domination problem is fixed-parameter tractable for d-degenerate graphs when parameterized by the size of the minus-dominating set and by d. The minus-domination problem is polynomial for graphs of bounded rankwidth and for strongly chordal graphs. It is NP-complete for splitgraphs. Unless P=NP there is no fixed-parameter algorithm for minus-domination. 79,1 5%

Edge-clique covers of the tensor product

In this paper we study the edge-clique cover number, ? e ( ? ) , of the tensor product K n × K n . We derive an easy lowerbound for the edge-clique number of graphs in general. We prove that, when n is prime ? e ( K n × K n ) matches the lowerbound. Moreover, we prove that ? e ( K n × K n ) matches the lowerbound if and only if a projective plane of order n exists. We also show an easy upperbound for ? e ( K n × K n ) in general, and give its limiting value when the Riemann hypothesis is true. Finally, we generalize our work to study the edge-clique cover number of the higher-dimensional tensor product K n × K n × ? × K n .

Results on Independent Sets in Categorical Products of Graphs, the Ultimate Categorical Independence Ratio and the Ultimate Categorical Independent Domination Ratio

We first present polynomial algorithms to compute maximum independent sets in the categorical products of two cographs or two splitgraphs, respectively. Then we prove that computing the independent set of the categorical product of a planar graph of maximal degree three and K 4 is NP-complete. The ultimate categorical independence ratio of a graph G is defined as lim k → ∞ α(G k )/n k . The ultimate categorical independence ratio can be computed in polynomial time for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. Also, we present an O ∗ (3 n/3) exact, exponential algorithm for the ultimate categorical independence ratio of general graphs. We further present a PTAS for the ultimate categorical independence ratio of planar graphs. Lastly, we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).