The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights

Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over the Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric. Holant problems are indeed read-twice #CSPs. Intuitively, some #CSPs that are #P-hard become tractable when restricted to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set F satisfies the condition, then F is of affine type or product type. Otherwise (a) Holant(F) is #P-hard; or (b) every function in F is a tensor product of functions of arity at most 2; or (c) F is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability

Favoring Eagerness for Remaining Items: Designing Efficient, Fair, and Strategyproof Mechanisms

In the assignment problem, the goal is to assign indivisible items to agents who have ordinal preferences, efficiently and fairly, in a strategyproof manner. In practice, first-choice maximality, i.e., assigning a maximal number of agents their top items, is often identified as an important efficiency criterion and measure of agents' satisfaction. In this paper, we propose a natural and intuitive efficiency property, favoring-eagerness-for-remaining-items (FERI), which requires that each item is allocated to an agent who ranks it highest among remaining items, thereby implying first-choice maximality. Using FERI as a heuristic, we design mechanisms that satisfy ex-post or ex-ante variants of FERI together with combinations of other desirable properties of efficiency (Pareto-efficiency), fairness (strong equal treatment of equals and sd-weak-envy-freeness), and strategyproofness (sd-weak-strategyproofness). We also explore the limits of FERI mechanisms in providing stronger efficiency, fairness, or strategyproofness guarantees through impossibility results

Multi-type Resource Allocation with Partial Preferences

We propose multi-type probabilistic serial (MPS) and multi-type random priority (MRP) as extensions of the well known PS and RP mechanisms to the multi-type resource allocation problem (MTRA) with partial preferences. In our setting, there are multiple types of divisible items, and a group of agents who have partial order preferences over bundles consisting of one item of each type. We show that for the unrestricted domain of partial order preferences, no mechanism satisfies both sd-efficiency and sd-envy-freeness. Notwithstanding this impossibility result, our main message is positive: When agents' preferences are represented by acyclic CP-nets, MPS satisfies sd-efficiency, sd-envy-freeness, ordinal fairness, and upper invariance, while MRP satisfies ex-post-efficiency, sd-strategy-proofness, and upper invariance, recovering the properties of PS and RP

A Complete Axiom System for Propositional Interval Temporal Logic with Infinite Time

Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning

Approximation algorithm for maximum edge coloring

AbstractWe propose a polynomial time approximation algorithm for a novel maximum edge coloring problem which arises from wireless mesh networks [Ashish Raniwala, Tzi-cker Chiueh, Architecture and algorithms for an IEEE 802.11-based multi-channel wireless mesh network, in: INFOCOM 2005, pp. 2223–2234; Ashish Raniwala, Kartik Gopalan, Tzi-cker Chiueh, Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks, Mobile Comput. Commun. Rev. 8 (2) (2004) 50–65]. The problem is to color all the edges in a graph with maximum number of colors under the following q-Constraint: for every vertex in the graph, all the edges incident to it are colored with no more than q (q∈Z,q≥2) colors. We show that the algorithm is a 2-approximation for the case q=2 and a (1+4q−23q2−5q+2)-approximation for the case q>2 respectively. The case q=2 is of great importance in practice. For complete graphs and trees, polynomial time accurate algorithms are found for them when q=2. The approximation algorithm gives a feasible solution to channel assignment in multi-channel wireless mesh networks